First dig a ditch around the outside of the runs (a couple yards away from the fence, preferably, so you're not undermining it), with an additional part of the ditch leading the collected water away to lower ground.
Then make sure the coop itself is not dumping water into the run -- install gutters and downspouts if you don't already have 'em, and watch where the downspout water goes.
Gravel or sand or a sand/gravel mix WILL help a lot (we're talking a significant amount, here, like 3-6"). HOWEVER it won't help for long if you put it in now when the ground is muddy. Sand or gravel dumped onto muddy ground will just sink into the mud and disappear, within months or a year, wasting all that money and work. Ideally, put gravel in when the ground is firmly dry. If gravel and labor are cheap you *could* put some in now as a stopgap measure, but be aware you're almost certainly gonna have to add a bunch more next summer.
An alternative stopgap measure would be to get a bunch of shredded bark or tree chippings or like that, preferably coarser stuff rather than finer. It will gradually start to decompose, and as it breaks down it will start making the mud WORSE, so you need to make real sure to rake/shovel it all out of there before then... but for a short-term solution, it can help keep the birds up above the water level.
Good luck,
Pat
