He knows 'fetch' and 'leave it'. We found he was more responsive to 'leave it' than to 'give' or 'drop it'.
When we want him to stop doing something - stop barking or whatever - we tell him 'enough'.
Never done it myself, but I know how I would do it.
Start by rewarding "drop"(leave it, whatever) when he does it exactly on cue.
Grab the basket you want him to use, place it under his head give him the "put 'em up" command, and have him "drop" it while you keep the basket underneath. When he gets it in the basket, praise. Then start moving basket, a few inches at first, praise when he gets it in, none when he "misses". Then put the basket where it will stay, and have fun with it. Play fetch and instead of bringing them to you, have him put them away. Make it a game.
That way, he doesn't always confuse drop with "put 'em up".
Wolf-Kim has a good idea. Then you can ask him to bring some of them to you and put some of them up and then get the toys from you and put them up, too. Sounds like you could have all kinds of fun with it!
I read about a person with an Akita (of all things) that new the name of each of 50 toys and would fetch the right toy when asked for by name. The owner would hide them and the dog would have to look for them to fetch them. I bet Jax would like something like that once he learns! A mental challenge for the genius dog