What does CERN do with its electrons?

You're right that CERN gets its protons by ionizing matter and collecting them. But the number of electrons & protons CERN deals with is far smaller than you might think. They get about 600 million collisions a second at CERN. So call it 1.2 billion protons used per second. $1.2 \times 10^9$. That'd be a large number in dollars, but it's not much in Coulombs.

For comparison, a wire carrying a 30 amp current has about $2 \times 10^{19}$ electrons flowing through it every second. That's a factor of 10 billion. So there's not really any issue disposing of CERN's unneeded electrons. You probably make a bigger spark when you rub your feet on the carpet.

If memory serves, the LHC has been running for its whole history off of a single canister of hydrogen gas.

The usual thing for a shutdown is to 1) stop injecting fresh particles into the beam tube, and 2) deflect any remaining particles in the main tube and any storage rings into a beam dump which is a very large chunk of metal, a very very large chunk of concrete, or a very very very large pile of earth. Take care not to be standing next to the beam dump- the radiation it produces while stopping the beam will kill you.

If your beam is working with electrons, you make them by stripping them off a hot piece of metal or ionizing you some hydrogen. In this case you steer the unwanted protons out of the resulting beam and run them into a dump. There they will find themselves some loose electrons lying about and get happy again.

Just to add an electrical engineering answer to these good physics answers, insulators are never perfect. In school we talk about perfect insulators, but in practice everything has some level of conductivity. Air itself has a resistivity of somewhere on the order of $10^{16}\Omega-m$, for example.

Usually when doing a process that generates ions is done, one side is "grounded," which means we let the charge flow into the ground. The earth can take an enoremously large number of these electrons before the electrostatic forces between them start to add up. And that gives time for these electrons to re-pair with protons.

Failing that, combining the electrical charges into the earth means our effect is combined with the electrostatic effects at the planetwide level. For example, the aurora borealis is a product of a massive stream of charged particles coming from the sun at a magnitude most of us cannot even fathom!