This is possible but from Basel to the closest "real" mountain (Alps) are at least 2 hrs.
If you are not fussed about snowy peaks and rocky terrain then I would highly recommend taking them walking in the Jura hills (Delemont, St. Ursanne...) or in the hills between Liestal and Olten (Schoental, Oberdorf). These are both within 1hr of Basel main station.
If on the other hand what you are really looking for is cable cars, peaks, snow, and so on... then the best bet is probably a fast train to Luzern and then a visit to Titlis, Rigi, or Pilatus.
One last suggestion that I have done with my godkids is to go to the caves at Beatenberg. This is a 10min bus ride from Interlaken (2hrs from Basel) so plenty of beautiful mountain views enroute, and then some walking at the caves to get to the entrance, and then about a 45min tour inside the caves deep into the mountain. The kids found it cool (and yes it was also very cold inside), and there is also a playground and piknik area there as well.
Either Luzern+ or Interlaken+ will turn into a very long day with two kids - so be sure to look up the return transport schedules before you go.
Hope it helps.