I stayed in Juneau a couple of days and started early to head to Mendenhall on the City Bus. The bus is $2 each way and takes maybe a half hour or so to get to the drop off point, which then is an easy and pretty mile and a half walk on a paved trail to the visitor center area.
The trail to the falls is another mile walk, then you walk it all back, but if you are in reasonable shape and have good shoes you should be okay. Have some water and snacks secured in a bag.
I also brought a small portable, foldable camp stool/seat in my bag that I used at the falls, so I could sit and look and enjoy in a bit more comfort, which I recommend.
When I arrived at the visitor center, there was hardly anyone, on my way back, there were 5 buses and more on the way, and the trails and overlooks were filling up, so go early!