How do I keep my folding bed side rail up?

You say that 3 sides fold up, but you didn't show the detail of how the end/side interface works. However, I'm thinking that some sort of decorative hook & clasp might work to hold them in the "up" position.

Something like this:

^{Image courtesy of Lowes.com. No recommendation of vendor or individual product implied or intended. For search term purposes, they call it a "gate hook and eye".}

You would attach the hook to the edge one piece and the loop to the edge of the other, then fold up one corner and hook them. Fold up the other corner and hook them, as well, and all 3 sides should stay up.

Of course, you also don't indicate what will be happening when they're up (using it to keep sleeping toddlers from rolling off?), so this may not be secure enough for your purposes, but it will hold. Of course, you don't indicate if you want a hidden solution, which this would not be.

I used a similar one, though smaller and made of brass, to hold the doors closed on a display case I made a few months back. It was quite suitable for the light duty I had in mind, this one appears to be much sturdier and may be more suitable for your purposes.

Use your favourite web client to search for terms like: click-clack hinge, sofa bed mechanism, etc.

There are many variations on these, but the idea is that they are a hinged locking mechanism with two (or more) resting positions.

Whether or not you can find a set that match your aesthetics is up to you. There are some that are meant to be exposed.

Alternatively, you could make a set of simple wooden "locks", either on a pivot (i.e., like a barn door or outhouse "lock") or by passing a small beam through two metal or wood brackets. The latter is sort of like those door braces you see in movies depicting medieval castles, which they close to keep out the marauding hordes.