Which desert? Among the biggest problems I have are the effects of cactus spines, and that varies hugely with the places I go. Sometimes I wear well-ventilated trail-runners, if I don't expect to hit many sharp plants; else I wear shoes with beefy toe and side rands. If the area had teddy bear cholla, take tweezers, an afro pick, and maybe even a pair of needle-nose pliers.
For slickrock I really like my La Sportiva Raptors. Grippy rubber and enough tread for the sandy areas, especially if you're talking about Moab area stuff. I put 50 miles on mine in the Needles area of Canyonlands week before last.
I'll probably be back in the area this week again.
I wear low top hiking shoe/boots. They're beefy, but not as much as a full-on boot. I like the low weight and feel. Leather gaiters may be wise if you think you'll be poking through cactus gardens or where snakes live. Always wear long pants.
What MoapaPk said. Deserts come in all sorts of flavors. I too carry a multitool and the pliers are useful in yanking out those nasty cholla barbs. Cholla balls can go right through most shoe material, except for thicker leather shoes. But it's a trade-off of weight and comfort. If you get a cholla stuck to you, don't make the mistake of grabbing at it to pull it off. Bat at it with a stick to get the main ball off, then start pulling out the barbs one by one. In time, your skin gets thick with scars and you don't feel them any more.
I got the afro pick idea from a friend who takes one on all hikes with potential cacti.
"Desert" means different things to different people. The Sierra Club Desert Peaks Section includes such diverse summits as Ruby Dome in Nevada (which has a tarn, and snow in normal summers), and the Guardian Angels in Utah (on which I saw relatively few cacti, and took sticky-rubber approach shoes), to Chemeheuvi and Mitchell Point in California (which are cactus-fests).
I try to avoid rattlesnake season, but have certainly seen them in Zion in April. Normally I'm moving slow enough that I hear the rattle before I get too close. I've investigated snake-proof gaiters, and have never convinced myself that I could be comfortable wearing them... there are some that sure could cut down on leg damage from spiky plants as well.
The problems with "jumping cholla" usually happen indirectly. You unknowingly step on the cholla, and a piece embeds in your sole and the side of your sole (particularly on shoes that have exposed foam midsoles). The you raise that foot that has the cholla in the sole and the protruding parts of the cactus hit the ankle and calf of the other leg, where they re-embed.