The Denver Botanic Gardens is not exactly known for being generous with its free days. If it's not doing them in the dark, it does them on weekdays when most people are at work -- probably the closest you're going to get to a free day on the weekend is at the DBG's Chatfield location, which is offering free days on the first Friday of every month until March. Still, if you can get there, you can bet it won't be crowded.
In opposition to the gardens on York Street, the gardens at Chatfield are more open and less manicured, more focused about the natural grasslands and ponds of deer creek than reflecting pools and sculptures. But it's a beautiful place for a hike if you don't have to pay for it -- which you don't. Winter hours are 9 a.m. to 5 p.m.
If Mondays somehow work out better for you, there's also a free day at the DBG's York Street location on January 17.