Budget and a few others have rental lots downtown Denver. I've flow into DEN, rented a car at the Denver Airport, and dropped it off downtown Denver if I'm riding the rails out of town. I've walked from the Budget lot to the station. Not a killer walk if you are in reasonable shape, about 8 city blocks. I usually stay at the Oxford Hotel the night before if you are taking the west bound in the AM.
Although I'd totally agree with warbonnet normally, with the relocation to the new temporary Denver station, it probably will be a bit more of a walk. I remember there was a budget city lot about 5 - 10 minutes from the station, so that went well and they did pick us up when we arrived. However, if you drop in off hours (Sunday in particular they're closed) walking could be a bit more of a workout. I'd think you could call a taxi downtown, though, and it'd be manageable.
Maybe with all the alignment of transportation services at the new station, some rental car service will move to Denver Union Station in the future? One can only hope.