I have done two things, although, I don't know if they'd help with strawberries. I carry my office clothes in a back pack, which bungee to my rear rack on the two bikes with racks, and wear on the road bike. When I have smaller or more delicate items, I put them in the back pack and wear it.

On my MTB based commuter I also have Wald folding baskets. I have laced a bungie cord from side to side above the basket bottom to provide a springy base. It can't be too taught or the base will fold up. I suppose this would work better in a non-folding basket.

I haven't tried this next idea, but it may work. If you find yourself without a basket or backpack I would try to find a few plastic bags fto make a remporary back-pack. First I would double or triple bag the produce to ensure it won't breach the bag. Then I'd loop the handle of a new bag through the handle of the main bag on each side to function as straps that you can put each arm through and wear.

If you have a re-usable canvas tote bag you might try handing it on your shoulder like a purse. Or if the straps are long enough, across you shoulders like a messenger bag.