Just allow multiple resolutions. For starters, always rendering the game in 1920x1080 on a computer with a monitor that doesn't support that resolution would be a waste of processing power. Secondly, it's a wise thing to do in case the game doesn't run well on older computer.
The key for this to work well is to make sure that your graphics are dynamically scaled proportionally to the chosen screen resolution. For example, if you're designing all of your graphics for 1920x1080, then the scaling factor for your graphics would be:
DisplayWidth / 1920
You'd apply this scaling factor(multiplied by 100) to the display zoom level. This will ensure that all graphics, when displayed at full screen regardless of resolution, appear the same size.
I hope that helps. If you want, I could whip up a quick example later tonight.