I'm heading to Geneva for a business trip, but I'm having some trouble
booking hotels.  I basically need either three hotel rooms (or an
apartment with two bedrooms and a living room) with check in-out dates
of April 9 & April 15.  The difficult bit is that I'd like to spend
less than $380 USD per night.  We'll be going to meetings near the
central train station, so anything within a 20 minute cab ride is fine
with me.  Two stars or higher would be ideal, but short term apartment
rentals are more than alright as well.  If this is possible, can
someone show me how?  Thanks!

Why not try the "Hotel de Berne". I haven't been there in years, since
I no longer  have to go to Geneva, but it was always very reliable and
is only 10 mins walk from the railway station.
26 Rue de Berne, Geneva, 1201 Switzerland
from $202.60-$254.33 (CHF 235.00-CHF 295.00)/night

That's a bit of a tough one. But after quite some digging, found a
site that hope will help. Copy and Paste the following URL in your web
browser:

http://www.precisionreservations.com/Afph/Afredirect.asp?urlToGo=http://www.precisionreservations.com/ph-aspx-city_id.16592-db-hoteldatabase/PHCOther.htm&cid=23285

If you change the "Show all rates" to the $75-$124 range, there should
be 4 options. When I checked, the Drake Langchamp had rooms available
for the dates you mentioned. I didn't check the others.

Hope this helps. Good luck! Err... sorry. "Bon chance!"