I use my larger air compressor in the garage for airbrushing, and the only thing you really need to worry about is which threading standard your equipment uses. Most equipment with be listed as NPT, NPS, or BSP. NPT and NPS are arguably compatible, but BSP and NPT will not mesh. These are the standardized thread patterns for pipes, and make it easy to ensure your stuff will work together. Most airbrush parts use either 1/4"|1/8" NPT or BSP. Some airbrush manufacturers use a proprietary/obscure fitting, but that's just a case of getting that manufacturers fitting.
So with my setup, I use a paasche H for all my airbrushing, and a lightweight bluehawk air compressor that I bought for nailing. I have a 50 foot rubber hose connected to the compressor (for no real reason, I just didn't want to take it off), and then the moisture trap/regulator, and finally my paasche hose. The moisture trap/regulator, and all the hoses are fitted with quick connect fittings to make it really easy to connect and disconnect things.
www.amazon.com/.../B003DN7716
This is an example of what the quick connect fittings look like. These are specifically threaded to connect to 1/4" NPT fittings.
I don't know what fitting sizes badger hoses actually use.
*EDIT* After taking a look at that air compressor, I did notice that it looks to come equipped with a female quick connect fitting. So if that is the case, you may only need to get a male quick connect fitting to connect your airbrush to it.