Connect MacBook to External Monitor Using USB to HDMI
A 1080 x 1920 display is easily within the capabilities of your MacBook Air. So, there are really only three possible reasons why this isn't working:
1. The monitor itself doesn't work.
Is this a brand new display, out of the box? Does it display a 'not connected' or 'no input' symbol on the screen, so that you can see that the screen itself works? Can you test it with another computer?
2. The MacBook Air doesn't work
Can you test the MBA with another display? Does the internal display work fine? Have you any reason to suspect that it might not work? Does the port work with other USB devices?
3. The cable/adaptor doesn't work
This is by far the most likely explanation, and the cheapest to test. If you have a USB to HDMI adaptor, followed by an HDMI to HDMI cable, try changing the cable first. Then try changing the adaptor.