Is there a bunny hill or seeing the Alps in the Basel part of Switzerland?

Not sure what a "bunny hill" is (apart form a hill with lots of rabbits).
First question, what method of transport will you be using, car or train/bus?

Alsace is a flat valley, running south-north. On the "left" side is the Vosges (hills/mountains). On the "right" side, across the Rhine is the Black Forest. The foothills either side are wine country, with pretty villages, French Alsation wine or German Baden wine.
If you can get high up in either the Vosges or Black Forest you can see over the Jura (foothills south of Basel) to the Alps beyond.
This map sort-of explains it: https://upload.wikimedia.org/wikipedia/en/7/72/Rhinegraben_sat.jpg

The highest point in the Vosges is the "Ballon d'Alsace", and you can definitely see the Alps from there, but you need a car to get up there. Website: http://www.ballondalsace.fr/
Blauen in the Black Forest is also an option. Wikipedia: https://en.wikipedia.org/wiki/Blauen_(Badenweiler)

Another option, if you have a car, is to go up into the Jura hills south of Basel. The Passwang Pass is famous for Alp views. Info (in German): https://www.baselferien.ch/ausflugsziele/passwang-aussichtspunkt/

One place near Colmar your wife would not like, but I recommend to anybody else is Château du Haut-Kœnigsbourg. It is above the vineyards with views across the plain.
Website: https://www.haut-koenigsbourg.fr/en

Basel is pretty flat countryside along the Rhine. You'll need to head further south and east to start seeing mountains. We ended a Rhine cruise there that included a day trip to Lucerne - about an hours drive away. Lovely town with some nice views of the mountains and lake. Pull up a geographical map of Switzerland and you'll see some other options like Zurich.

P.S. Lindau is a pretty town on the German side of Bodense (Lake Constance) which is worth a stop and has views across the lake to the Alps.

No, you'd have to go much further into Switzerland to see the Alps. Not practical from Colmar, you're better off enjoying the nearby villages and hills.
But regarding fear of heights: you can visit the Alps without leaving the ground and without standing at cliff edges. There are plenty of scenic rail routes in Switzerland for instance - maybe for another trip?

A "bunny hill" is a term sometimes used when referring to a smaller ski slope made for kids or novice skiers.

It is about 2 hours by train from Colmar to Lucerne (Luzern). The city is at the north end of the Lake of Luzern (Vierwaldstattersee), which has spectacular lake cruises all the way to Altdorf (home town of William Tell) at the south end in the middle of the front range of the Alps. You won't get closer any more easily.