At first this questions tends to get the obvious answer: practice! That is, repeatedly engage and disengage your feet from the pedals. Don't go for a normal ride, ride a few meters and stop, then start again. repeat over and over until you brain forgets about it because you have built muscular memory.
Yet another exercise might be this: At home, get yourself on the bike inside a door frame. Now, while pressing the brakes and with one foot already engaged, hold your balance and quickly lift your other foot from the ground, clip, unclip and return it to the floor. If you fail, just grab the door frame before hitting the ground. Repeat over and over, and then do the same with the other foot. This will help build muscular memory as well as force you to do it quickly.
Another exercise is, an empty parking lot, or an ample flat terrain, try to make balance while riding as slowly as possible without removing any foot from your pedals. You can even get to a complete stop and remain clipped. This stunt is known as a trackstand. This will improve balance and teach your brain to stay on the bike, instead of thinking about putting a foot on the ground as a first resource.
Those are practices, but when you actually have to stop and unclip, critical fractions of a second prevent you from doing it on time and you crash. When you traverse a technical section, take a previous decision on what foot to put down, that way your brain will have one less task to do in those critical seconds, freeing it to drive that foot effectively when needed.
That's the answer for the person that just bought and (got) installed clipless pedals for the first time. But I'm pretty sure any rider would feel better by actually conquering those sections without putting any foot down (regardless of pedal type). Effectively improving your climbing techniques will minimize your need to repeatedly put a foot down, and will minimize the chances of a fall because of that reason. Rather than fearing a fall, focus on techniques to overcome the obstacles you are facing, concentrate on pedaling and steering, be as conscious as possible when selecting the line you ride.
The question specifically expresses concern for rear wheel slippage. There is a couple of things that can be done to minimize that.
Learn to control where your rear wheel goes. Remember that except when going straight, the rear wheel follows a different line than the front, but at an almost fixed distance, so through exercise you can get to the point where you always know exactly where it touches the ground. I recommend practicing on a flat ground, like an empty parking lot or playground. Dispose makeshift "transit cones" in line, separated a regular distance (about the axle to axle distance of your bike) and practice zigzagging the rear wheel without touching the cones. Then vary the distance between cones and make different lines. Once you gain control on where the rear wheel goes, you'll be able to pick better lines in your technical climbs.
Choose a gear that allows you to remain seated. This puts more weight on the rear wheel and gives more traction. It also demands less power from you, you will be less tense and since you'll be a little slower, leave you a little more time to select the line
If possible, use a little lower air pressure on the rear tire, as it increases traction too. However, don't lower it too much because it also increases rolling resistance and increase the possibility of getting pinch flats.
A correct posture will also help a lot. While seated slide a little towards the front, lean your torso towards the handlebar, firmly grasp the handlebar with your forearms parallel to each other and to the ground. This position gives a more solid weight distribution which gives better traction, allow for more precise line selection an better power transfer, but it also gives a better chance to recover from a slippage without putting any foot down.
Distribute your power. Sometimes a climb has short sections that are steeper, but the general gradient is more tolerable. If not steeper, there are short sections with more difficult terrain, while the rest is smoother. On the smoother or less step sections pedal a little below your full potential (think of it like "pedaling at 70%"). Accelerate to full power two or three pedal strokes before the steeper or difficult section, but don't "explode" into full, instead smoothly and gradually accelerate during those few strokes. Once you reach the section at full power, concentrate on pedaling, keep the rhythm (cadence) as constant as possible throughout the section, and resume 70% percent when you get o a smooth or less inclined part. During that short burst try to follow the line you must have selected before entering the section. This "70%" thing lets you save energy for the rough parts of the trail, and let you recuperate your breath after them.
Do not shift gear while in the roughest parts of the climb. A badly executed gear shift is as bad as rear wheel slippage, but it may compromise the transmission. Also, as a general rule, shifts should be avoided when pedaling at full power. I recommend doing the opposite, "soft pedaling" while the shift occurs, this can't be done in the roughest sections.
I also recommend reading these answers: Bailing out with SPD pedals on a mountain bike
Besides that: Patience, persistence, perseverance and progressiveness.