Are "rusty" capacitors enough to cause a computer monitor to malfunction?
Those caps look like they’re failing. Notice that the ends are bulged out - the score marks on the caps provide a relief point for them to ‘vent’. And it looks like they’re doing exactly that. That ‘rust’ is electrolyte being expelled from the cap.

What to do then? Stop using this monitor - the caps may eventually short out and catch fire.

Now, as far as fixing it...

The placement of the caps looks like they’re on the power input side (DC in). That’s suspicious. It leads me to believe that their voltage was exceeded. Is the AC adaptor misbehaving? If so, more things may be fried than just the caps.

If you want to try to repair the board, make sure you understand that electrolytic caps come in different types. Try to find the exact type and cross that for replacement. In particular, note that low-ESR caps are used in power supplies due to the high dynamic currents they must deal with.

And, given the age of the monitor, it may have fallen victim to the ‘capacitor plague’ of the early 2000’s. More about that here:  https://www.theguardian.com/technology/blog/2010/jun/29/dell-problems-capacitors

Even then, check the AC adaptor to make sure it is working correctly. Test it with a load to ensure its voltage is in spec. It, too, may have the wonky caps.

At the end of the day, if it’s an old monitor it may not be worth the repair.