On a scale of 1 to 10, probably about a 7. In other words, not earth-shatteringly important but still pretty important.
A Yagi-style antenna like the one in the picture is usually mounted level when you put it up, but over the years it can droop a little bit. This generally isn’t too much of an issue because of the design of the antenna itself. The yagi antenna isn’t really one antenna but many; each one of those horizontal elements is a specific antenna for a specific range and they are designed not to interfere with each other. They still won’t interfere with each other if the antenna’s a little droopy. However, if it’s a VHF/UHF antenna (and most are) then the UHF antenna on the front is going to get less effective as it droops over.
In fact the smaller UHF antennas are going to be more of a problem if they droop over, simply because of the design of most UHF antennas. Loop or bow-tie designs get a lot less effective if they’re not facing the towers and if they’re pointing downward they’re certainly going to lose effectiveness. This is true of any antenna — it must be pointing toward the source of the signal — but UHF antennas are more sensitive to this.
And of course, I’ll point out that a droopy antenna is unsightly and I suspect your neighbors would appreciate it if you fixed it. Personally I see a lot of droopy antennas in my neighborhood, owing to the fact that the antennas themselves have been up there on the roof for fifty years or more. People may not even be using them but may not know how to safely dismantle them. It’s usually just a matter of loosening some bolts if you need to remove the antenna, but it’s usually just as easy to tighten those bolts if you would rather keep an antenna on the roof and start getting free TV.