Ballard is 5 miles from downtown so it is not a bad trip by bus at all. It has a large number of indie restaurants (including several of the city's hot restaurants with more on the way), bars (including some notable beer and cocktail bars) and has several great live music venues (most notably the Tractor, the Sunset and Conor Byrne) -- largely centered on historic turn of the 20th C. Ballard Avenue, one of Seattle's hot nightlife areas, and adjoining Market Street and Leary. There are also currently 8 breweries in Ballard with two more on the way plus a number of small art studios, boutiques, many indie coffee shops, patisseries (particularly outstanding Cafe Besalu), good book and record stores, a well regarded indie movie theater, a small stage, etc. Ballard, unlike Capitol Hill or Queen Anne has Puget Sound and ship canal waterfront. There is a sandy beach at Golden Gardens where it is possible to see intertidal life and just in the last day or two a seal was spotted on the boat launch sunning. Nearby the locks, which connect Puget Sound with the ship canal, lake union and lake washington, are a big Seattle attraction. The grounds have a small botanical garden, a free museum and visitor center and a fish ladder that returning salmon and steelhead use to bypass the locks. You can also spot harbor seals, sealions, blue heron and eagles at the locks (although only the blue heron are mostly a sure thing). There is also a neat local museum, the Nordic Heritage Museum, celebrating Ballard's Scandinavian roots and on Sundays Ballard Avenue hosts perhaps the biggest year round weekly farmer's market in the city (obviously apart from permanent Pike Place Market) with farmer, forager, food artisan and flower stalls, prepared foods and musicians busking at various places around the periphery. Ballard has a very unique Scandinavian and maritime feel which I have posted about a number of times and I consider it one of Seattle's most interesting neighborhoods.
Ballard also has the advantage of being connected by a very convenient bus route with Fremont, another of my favorite neighborhoods which is also heavy on indie character, public art, bars, restaurants (which has had a streak of hot restaurant openings) live music venues, and more. Fremont is a bit closer to downtown (an easy 3 miles approximately) and convenient to Seattle Center, South Lake Union, Ballard and the University District. It is one of the neatest and quirkiest of Seattle neighborhoods and has a great feel with some of my favorite restaurants (Revel, Joule, Whale Wins, Agrodolce), bars (Brouwer's - perhaps the most exciting and influential beer bar in the city and Six Gill), one of my favorite breweries (the outstanding Fremont Brewing Co.) and good live music options (High Dive, White Rabbit, Nectar, Fremont Abby Arts Center).
Capitol Hill and Upper Queen Anne hill (like Ballard and Fremont) are safe. Queen Anne is the most upscale and quiet. However upper Queen Anne has a very nice hill top stretch running north south along Queen Anne Ave. There are many restaurant options including a couple very good ones like Lloyd Martin and How to Cook a Wolf plus some budget crowd favorites like Mescaleria de Oaxaca (sister restaurant to Ballard's La Carta de Oaxaca). There is also a great patisserie (La Reve), several good coffee shops, and a number of small indie boutiques. Just a few blocks down the hill is the excellent Kerry Park overlook of downtown. it is one of the iconic views of the city with the Space Needle in the foreground and the city and Mt. Rainier in the background (basically the Frasier view). It is walkable to Seattle Center but I would caution that the uphill return trip is a very steep hike. There are also no hotels on upper Queen Anne. There may be a few small B&Bs. I would have to research that.
In general Ballard, Fremont, Queen Anne and most of the desirable north and NW neighborhoods do not have a lot of hotel options. Ballard, which is developing rapidly, has a small European style inn and a brand new boutique hotel on Ballard Avenue. Otherwise there are just B&B options. I would carefully consider how close they are to the center of Ballard as Ballard is very large and hilly. Fremont has one hostel and B&B options.
Capitol Hill's hot area is mostly along the Pike and Pine corridors (particularly East of Broadway_, the area along the East side of the Seattle University Campus, Broadway, the area just east of I-5 between Melrose Market and Olive and up Olive. This part of the neighborhood is densely packed with apartments and condos. It is much more urban than the top of Queen Anne. There are a few hotels, mostly at the Southern fringe near Seattle University and First Hill. There are some very nice B&B's that are clustered closer to Volunteer park but you may find some other B&B's throughout Capitol Hill via services like Airbnb.
Capitol Hill has a disproportionate number of restaurants from our nationally lauded and award winning chefs. They also have a number of the city's best bars, close access to the city, indie boutiques, live music venues, coffee shops, patisseries, indie theater and film spots and one of the city's best bookstores Elliott Bay bookstore. It is the heart of the city's gay community and also arguably the most urbane neighborhood. It is safe but like all areas of town in which there is nightlife with people on the streets late (including Ballard and Fremont) I would exercise some situational awareness if you are out very late.
In short the urban center of Capitol Hill is very urbane. It is literary, artistic, young, gay and has lots of options for entertainment. It is very close to downtown. You could walk although heading back uphill to Capitol Hill (depending on the location) can be a bit tiring depending on the location of your B&B and your shape. It is served though by a number of bus routes which you can take if you don't feel up for the walk. It can be loud at nights depending on where you are. I consider it safe but there is some crime and it adjoins areas where crime could be more of a concern so if out late just use normal city precautions.
Queen Anne is next closest. It is particularly convenient to the Seattle Center part of downtown and the bottom of the southern slope of the hill (lower Queen Anne) has additional shops, bars, restaurants, theaters etc. The top of the hill around the commercial center is more affluent (Queen Anne has a few areas among the most affluent in the city). It has a number of options for entertainment but I would say overall it would be considered quiet by comparison to Capitol Hill, Ballard or Fremont. The hill is very steep so walking back and forth will likely not be your regular preference but walking down is not so bad and the top of the hill is well served by bus routes serving the hill, downtown and Seattle Center.
Fremont is next closest. You could walk downtown from Fremont but it would be a long walk. It is a fairly easy bike ride via trails but there is also excellent bus service making the trip quick and easy. Fremont has more character than Queen Anne and arguably has more of its own unique identity than Capitol Hill. It cannot match the number of eating and entertainment options of Capitol Hill but is an increasingly hot neighborhood with lots of options for eating, drinking and other entertainment and it gives you easy access also to other great N and NW neighborhoods and their attractions like Ballard, Phinney Ridge/the zoo, the University District with the Henry and Burke museums etc.
Ballard is furthest from downtown of the options discussed here. However it is linked to downtown and other neighborhoods by a range of bus routes. In my mind it has the most unique identity of the downtown neighborhoods (with perhaps the exception of Pioneer Square) as it was its own thriving maritime and lumber industrial city with a strong Scandinavian identity. It was annexed by Seattle over 100 years ago but retains a strong maritime and Scandinavian identity today. It doesn't have the range and depth of cultural amenities that downtown -- or even capitol hill -- does but it is has literary events, theater, arts, and one of the city's hot neighborhoods for eating, drinking and live music and it is perhaps the most notable brewery district in the city.