Most LCDs run at lower resolutions just fine (they have to for now, in order to support text mode stuff like the BIOS readouts/CMOS setup/etc.), but that's not all you have to be concerned with.
If you ever run into a problem wherein your monitor puts a strange border around the stretched image at lower resolutions (and only at lower ones, i.e., 640x480 and below) and you're using HDMI, try VGA. I had a monitor that was doing that to me (28" Hannspree) - took me forever to figure out a way around it.
The panel type plays a big part in color reproduction - an IPS or PVS panel produces superior color variety without dithering when compared to a TN panel (typical monitor). I always thought TN panels were fine, until I had an IPS sitting in front of me. It's so much closer to a CRT in color quality - but most DOS games are very low in number-of-colors, so depending on what you're doing this may be irrelevant.
In addition, the ratio of screen size to resolution (called pixel pitch, the equivalent of dot pitch in CRTs) is important. If you don't have good eyesight, don't worry about that, but if you do, the smaller the pixel pitch the better - at around 0.25 pixel pitch I can't see the pixels without being an inch or two from the screen, and I've got incredible vision.
A big advantage to emulating classic games - when using an LCD only, mind you - is that the emulator (in this case DOSBox) can scale the image up to the native resolution, or the PC's hardware can. Often one of those two choices produces a better image than a monitor's scaling - but that depends on the monitor and GPU. If you get a monitor that you like but you feel it would benefit greatly from some better scaling, there are boxes you can buy that are dedicated to upscaling images better than your typical device can (but these can be expensive, I've never shopped for one though, so there might be cheaper options).
Personally I purchased a Dell monitor with a glossy finish (anti-glare kills my eyes), 21.5" 1920x1080 (.248 pixel pitch), and an IPS panel, and I'll be using it for DOSBox and real DOS machines (though I always keep a nice CRT nearby, usually my 21" Sony E540). I also bought a Matrox G250, which will be capable of running that monitor at 1920x1080 and has drivers for Win3x & 9x as well as good DOS support - it also has drivers in much newer OSes at least upward of WinXP (which means those drivers can be forced in Vista/7 if they don't have their own built-in or available somewhere, not that I'll be running the card on those OSes). That'll go in my primary retro box so I can share the monitor with my modern box (at native resolution under 9x).

^- Monitor I ordered, a ST2220T, Dell's only glossy IPS - also happens to be multitouch, that'll be amusing.. 😁