Your baskets likely need two or more points of connection, which can be the top corners but in some cases you can use a series of points in a vertical line in the middle. With some very sturdy baskets you can get away with just one point of connection, when using a rigid frame, but do check for droop and twisting of the basket itself.
Once you have decided how you can or have to connect the baskets to your frame you can start thinking up what your frame will look like.
I am a knots person and my initial reaction is to use rope, connecting the baskets on their ends to the ropes.
Which will then allow you to compress it while off the door but expand and give plenty of access when on the door.
To make this kind of hanger you either need two ropes with a hook to attach it at the top of each, or easier, one rope folded double with an eye in the middle. The way they handle the rope in this rope ladder shows what I try to say.
If you feel more like a rigid frame, by all means make one.
You can make a single 'top down' frame, or you can make a few separate ones that you can hook into each other, adding more baskets when you need special items.
Or you can just use one basket which is big enough for your needs.
If you can attach a hook or nail to the door you can easily just use a loop or eye.
Otherwise you will have to add a hook that hooks over the top of the door. You can bend on from a flat piece of metal, as long as the metal is strong enough to hold the shape.
Or you can go for any of the 'over the top of the door' hooks you can find in shops.