Can "power sags" be harmful to a computer?

A quality PSU can help with these, but you really do want a quality UPS if this is a constant issue. If you are worried about outages, a pure sine wave UPS is ideal:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Or you can just get a good line conditioner:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

I think they do have some cost to run though... You burn some power.

A good surge protector will guard against dramatic issues, but not the more subtle ones that eat up electronics over time.

Is there any way to monitor if I'm getting power sags?

Dimming lights?

Really though, it depends on how closely you want to look, and how much you want to spend.

Here's about the equivalent of the lights thing only a bit fancier  

http://www.amazon.com/GenTran-Portable-Generator-Voltag...

I probably have one of those in my basement. I'm really just not sure if it's a problem or not. What kind of damage can power sags do?

http://en.wikipedia.org/wiki/Under-voltage_%28brownout%...

Get yourself a good UPS.
If power disturbances come frequently, then YES!, your PSU and/or your system will have shorter life.
Common power disturbances at home :
- permanent over or under voltage
- unstable voltage (up and down)
- bad grounding
- power spikes, either sudden drop or sudden high jump in a very short time.
Any of those are bad!
If you have any of those frequently, you must get a good UPS!