Joy, Yes, but.
Yes, Basel is a nice city, and is worth a day at least to explore. But, you only have two full days for the Alps. I don't know how late on the Sunday you are getting to Basel, but it is only 1 hour by train to Luzern. If you have a few hours explore Basel, but get a train to Luzern and wake up Monday morning and walk out to a view of the Alps.
The "past Interlaken further into the mountains" is Lauterbrunnen, Mürren, Wengen, Grindelwald etc. All of which are good places to start up the Jungfrau.
But but, you have a 10:45 flight, you need to get to the airport 2-3 hours beforehand. The first train out of Lauterbrunnen in the morning is 05:33, arrive Zürich airport 08:16. From Luzern it is (for example) 07:10 arrive 08:13, with plenty of earlier trains.
Same logic applies in the other direction. Basel to Luzern is 1 hour, Basel to Lauterbrunnen is 2½ hours.
This is why I recommended Luzern. You get your full 2 days in the Alps.
Options from Luzern are as I listed, a boat trip on the lake (you may be fed up with boats), go up a mountain (Titlis, Pilatus, search this forum or http://www.pilatus.ch/ and http://www.titlis.ch/en ) and lots more. Titlis will get you a (small) glacier and wonderful views.
More info on Luzern official tourist site: http://www.luzern.com/en/index.cfm
P.S. I'm not actually from Basel, but I have lived here for a number of years. I am actually English, spent about 3/4 of my life living there, with a lot of travels around Europe for work and pleasure, and ended up here.