Lasers.
You don't need to destroy the debris, just push it out of the way by the slightest amount. The vast majority of space junk is tracked, as others have mentioned, so you'll know what's a threat long before it actually hits. Hit the debris with a laser to alter it's path slightly and you can avoid collisions.
Because you can track most of the debris, you can even start this process well in advance. Debris will be speeding past very quickly, but you should be able to hit anything dangerous multiple times as it passes by with each orbit.
You can even use this system whenever it's not directly protecting the elevator to slowly reenter space debris as it passes. Just slow down the speed of whatever passes by and eventually stuff will start to fall out of orbit. It cleans up space and protects the elevator all at once!
As for the stuff that isn't tracked, you'll need to be able to detect it in advance, so some sort of sensing equipment will be necessary, but you'll need that already for laser targeting. Luckily the stuff that's not tracked is all very small stuff, so it will be easier to push out of the way, and you won't need as much advance notice.
Delving deeper!
Luckily for anyone who wants a thorough, scientific analysis of this option, I'm not highly original and laser reentry of space junk has been studied! One particular paper DOI: 10.1016/j.asr.2012.02.003 goes quite in-depth, and I'll cite some of its more important findings here.
First, how big of a problem is space junk?
There are about $N_1 = 2,200$ large objects (diameter ≥ 100cm, mass of order 1 ton) in LEO, and $N_2 = 190k$ small objects (diameter ≥ 1cm). 199
The flux for the small ones in the peak density region is about $R_2 = 1.4E-4$ $m^{-2}year^{-1}$ ... [Applying these figures], the chance that a big object will impact a big object is once in $T_{11} = 134$ years, whereas the chance a small object will impact a big object is once in $T_{21} = 3$ years.
One variable seems to change names at some point in their calculations, so I'm not certain about how this translates to a stationary object, but it does give us a time scale.
So if we do nothing, we'll have a small strike every few years. But let's not do nothing! Let's use some lasers!
First of all, how do lasers work to deorbit debris? Basically, the laser hits the surface and rapidly superheats the material, vaporizing it. This vapor is still in the path of the laser however, so it continues to be struck by the laser beam, where it superheats a bit more and becomes plasma. The plasma rapidly decompresses, pushing the object away, and essentially forming a small jet on the object's surface. After some time, adding more energy to the system is counterproductive so the authors of this paper suggest a pulsed laser, rather than a continuous one.
The authors assume a period of $\tau = 5ns$ for the laser pulse, and using this number, find that they need to apply $53\ kJ/m^2$ to the object to optimally accelerate an aluminum target.
In a practical case where $D_{eff} = 10m$, if $T = 80\%$, $T_{eff} = 0.5$. In order to deliver $53 kJ/m^2$ to a target at $1000km$ range, the product $ WD_{eff}^2 $ must be at least $993 kJm^2$, laser pulse energy must be $ 7.3kJ$, and the mirror diameter $ D$ must be $13m$.
This means that the lasers don't even have to be in space! With pretty much current tech, you can deploy a system of them on the surface and deorbit from there!
The authors find that any object with mass less than 1kg can be reentered in a single pass. Larger objects (their example gave mass of 1-ton) would take several years to reenter, but, if you recall the frequency of interactions between large objects, the chance of a catastrophic collision is very small, and we can very easily nudge these objects out of the way in a single pass. 580