Are you planning on coming for the full week or the 2 main days (wednesday & thursday?) If you are coming for just a couple of days - consider Friday and Saturday - most of the main visitors are gone and its mostly locals who attend on these days. Much easier to get to / from the racecourse and not as frantic!
As Harvey said - race week is oneo f the busiest weeks in Galway, even if numbers had dropped a little in the last couple of years and so the hotels tend to take full advantage of that with regards prices.
You have a couple of options - you can stay in Oranmore as already suggested, it has a number of pubs and restaurants so keep you entertained in the evenings.
You can stay in the city centre - there are plenty of buses going to and from the racecourse for easy access and you wont have to worry about getting taxis at night.
You can stay in Salthill - a couple of miles from the city centre, so you can still access the buses to the racecourse and it would be a manageble walk home at night if you could not get any taxis.
Where ever you decide to stay - book soon!