DIAGNOSIS You're right to be careful. The 'borderless' design means the very thin plastic bezel is essentially flush with the screen. Using liquid or pressure near the edge can seep underneath or damage the panel seal.
FIX Power the monitor off so you can see smudges better. Use a dry, soft microfiber cloth—like the kind for glasses. Gently wipe the screen in a circular motion. For stubborn spots, dampen a corner of the cloth with distilled water (not tap water) or a tiny bit of screen cleaner sprayed onto the cloth, never directly on the screen. Avoid the very outer millimeter where the screen meets the plastic frame.
IF THAT DIDN'T WORK For dust in the crevice where the screen meets the bezel, use a soft, dry makeup brush or a can of compressed air held upright. Tilt the monitor back and gently brush the dust out, don't try to wedge the cloth in there.
HEADS UP Never use window cleaner, acetone, alcohol, or other harsh chemicals. They can strip the anti-glare coating right off the screen, leaving permanent cloudy patches. If you get liquid in the edges, you might see dark spots that won't go away.