I live in the northeast and have a Desert King that is several years old and I get a nice crop of brebas each season. The tree is getting leggy; can I prune it after the figs ripen with the hope that it will still put on additional growth for next years early crop?

Hi,
Yes and no .
I doubt that where you will cut, that you'll get breba figs next year. You'll get brebas on the untouched stems.
So the more logical way to go, is to prune half of the tree this year, and the other half, the next year.
You could as well air the tree in the same way.
If you still have some place, you could airlayer some of the to be cut stems ( root them ), and install them in the garden to have more than on tree, and thus pruning the trees one at the time, one each following year.
Pruning is never easy, but at some point must be done.

Thats always been one of the questions and concerns I've had with Desert King or any other fig that primarily provides Breba figs.  Since the figs like to grow on last years wood and last years wood makes the best cuttings you are effectively reducing your plants ability to create figs every time you prune or take cuttings aren't you?  

I like JDSFrance's suggestion.  You could imagine having a Y-shaped tree with two main scaffold branches.  One year after fruiting you could prune back one side but leave the other unpruned.  Then then next year you do the opposite side, and so one.  That way half of the tree will be productive every year.  I haven't done this but on paper it sounds like a good idea.

An alternative may be to use a branch thinning strategy.  Each year select appropriate number of branches either suckers, criss-crossing branches, damaged or sick looking, or too close together and cut those out completely.
The remaining untouched branches will have more room to grow and should produce a lot of breba figs following year.

If you prune after picking the figs wouldn't the rest of this summers growth starting from where you cut it (the wood)  be what gives you next years breba?  So what's the problem?

I am thinking similar to Ed. I am going to prune my DKs after ripening of the few fruits they have. If it did not work I will give up on DKs altogether which I had been thing to do foe the past few years. I find pruning routine for DK a bit messy for my taste.