We sleep in the basement in the summer but have had issues with dampness. We just use twin mattresses from ikea.
I thought I saw something at ikea last year, like wire triangles that went with the wooden slats, but can't find it now
I made one for similar reasons. Three parallel pine sleepers with hardwood slats screwed across the top (basically a futon). The platform height is about 15cm (6 inches).
If you don't want to DIY, you might find a store bought futon is suitable.
2 of our 3 beds are homemade platforms - we went the plywood and 2x4's route. It's surprisingly easy - all the cuts you need can be made at the lumber store, so it only requires a drill and some long, sturdy screws - we used deck screws. For sturdier posts, we doubled up on 2x4's, held together with Gorilla glue, then screwed them to the platform using two screws.
The twin bed only needed 4 posts, but the king has 3 extras, forming a middle row - this may be overkill, but we have a tempurpedic mattress and the weight of it destroyed our previous Ikea bed, so we weren't taking any chances. Bonus: we were able to construct the bed to suit the height of the bins we wanted to store underneath.
Pros: cheap and sturdy, easy, fast (we built two in two hours, not counting the previous day's trip to the lumber store and overnight Gorilla glue setting)
Cons: we used raw wood and did nothing to finish it, but it doesn't show, so...I'd had some thoughts on wrapping the legs with rope for a decorative touch, but decided not to bother
So...I'm 43 and pregnant with #3 - how'd that happen?
I noticed that home depot sells precut wood slats for beds. So if you want to diy without too much diy-ing, you could buy those in the twin size length and just attach them to something. For instance, you could screw a series of bedslats on top of a 2X4 on each side. Then set it on something like milk crates or attach feet or legs, even just wheels, underneath to the 2X4s.