Hotel with AC in the black forest

Very unlikely. Air conditioning is not really needed north of the Alps (i.e. not anywhere Germany) and you are only likely to find it in big modern hotels that are so large you need some mechanical ventilation.
Staufen (full name "Staufen im Breisgau", click here for photo) is a small town of traditional buildings. The weather doesn't get that hot, I am not sure why you think you need Air Conditioning. The Black Forest is famous for trees and fresh air, Air Conditioning does not go with that ambience, if you need fresh air, just open a window.

Yes, Staufen is nice place to base yourself, on the edge of the Black Forest.
If you are looking on hotel websites, German for air conditioning is "Klimaanlage".

I took a quick look on HRS.com, searching for hotels in Staufen with air conditioning. It showed none. HRS is I believe the largest hotel search site. It did show some in Freiburg and its suburbs, one as near as 7 miles from Staufen. I have found the ratings on HRS to be accurate.

"Do you have better suggestions for a town to make our home base?"

What is better depends on what you plan to see and how you plan to get around. But I believe there are better options. The BF is huge. Staufen is on the southwest periphery. If you wish to locate yourselves more centrally, check out this map of the BF and target the Wolfach/Hausach area:

https://www.welt-atlas.de/datenbank/karten/karte-1-664.gif

Gutach (neighbor town of Hausach) is the home of the Vogtsbauernhof open-air Museum, a popular destination where you can get a glimpse of traditional life in the BF:
http://www.bensbauernhof.com/gutachimschwarzwald.html

Both towns are located on the scenic Black Forest Railway, one of Germany's nicest train rides. Triberg and Villingen lie on the same line and are also worth visiting. BF Railway map and info:

https://upload.wikimedia.org/wikipedia/commons/4/4d/Verlaufskarte_Schwarzwaldbahn_(Baden).png
https://www.black-forest-travel.com/places-of-interest/black-forest-railway.html

Nearby Gengenbach could be a good base town as well and far outshines Staufen on most measures, IMO:
http://www.der-ortenauer.de/ortenauer_bildarchiv/gengenbach/gengenbach_082008_022.jpg
http://www.reiseerlebnisfuehrer.de/wp-content/uploads/2014/04/Gengenbach-Engelgasse.jpg

Riding the BF Railway and the other trains within the BF is free of charge when you stay in certain towns that offer the KONUS card (which is free of charge) -

http://www.blackforest-tourism.com/info/KONUS

We stayed recently in Steinach, in between Gengenbach and Hausach, in an apartment at this farmhouse - lovely time:
https://www.traum-ferienwohnungen.de/13558/

Air conditioning is not really needed north of the Alps

Although I wouldn't make air conditioning a criteria in searching for a place to stay, it kind of depends on your definition of "needed". Although I survived a night in Rothenburg odT without A/C, it would have been really nice. Both the temp and % humidity were in the 80s one night. It was difficult to sleep with the windows open because we were on a busy street (the first one outside the pedestrian zone) and the car noise echoed off the walls of the 5-story canyon, so we roasted with the windows closed. However, the next night was fine. I've spent about 150 nights in Germany, and this was the only one time the temp was high enough to want A/C (that's less than 1% of the time).

I needed air conditioning in Erfurt, Quedlinburg, Berlin, Dresden and Munich between July 23 and August 11, 2015. I did not need it in Schwerin. The a/c was not operating in my Dresden hotel, and it was not a fun time.

But that summer was hot practically everywhere I went. I know it can be cool and rainy during German summers; it just wasn't for me. And I wouldn't have liked that, anyway.

bin,

I was in Freiburg im Breisgau in September, and at that time of year air conditioning would have been desirable. I imagine the situation would be worse in July. The hotel I chose did not have A/C and even with the windows open, the room was "stuffy" and sleeping would have been impossible. I asked for a fan, which was provided promptly, and that solved the problem.

If you're not set on staying in Staufen, Freiburg would also be a good choice. It's got a beautiful cathedral, lots of restaurants, good hotels, etc. The street market sets up around the cathedral on some days, and the odours from the various food vendors are almost irresistible!

The Black Forest area is the warmest area in Germany and it can get really really hot in summer! In Freiburg the temperature did go up to 42 C last year. Many private homes now have a AC unit because the temperatures have gone up and up over the past few years. If you are used to AC I highly recommend finding a hotel with AC. Just because you will be unhappy otherwise. Often hotel owners will tell you to open the windows at night to get some fresh air in. But - in summer there is often no fresh air - the air is warm and sticky. And Germans like to be outside in summer - so make sure to bring ear plugs if your hotel is located right in town:)
But - the good news - Many hotels have now upgraded to AC systems. If you go a bit more north - Freiburg for example - you will definitely find hotels with AC units. It also depends on the hotel category - the 4 and 5 star hotels in the Black Forest will most likely have AC, even if they are in smaller towns.

I lived about an hour further north of the terminus of the Black Forest, and I can concur, you can experience brief, but intense heat waves in SW Germany, particularly in the very broad plain of the upper Rhine valley. More than likely, temperatures will be comfortable, but if you get hit by one of these rare heat waves... there's almost no escape without air conditioning, except to head high into the hills.

Neither one of the Pensionen I stay at in Munich nor Berlin has A/C when I am there mostly in June and when it gets plainly hot. In Munich there might be a fan in the breakfast room, in Berlin nothing, ie no fan , A/C. Of course, the Munich Pension costs more than the one in Berlin. Basically, I don't need A/C, and certainly will not pay extra for it.

The Super8 in Freiburg im Breisgau has air conditioned rooms. I can't speak to its quality.