Hold down Command (the cloverleafkey) + tap spacebar to get Spotlight up. Type in 2 letters: sa. Safari will be highlighted in the resulting Spotlight dropdown list.
While holding down the Command key, click on the Safari *ICON* at the top of that dropdown list (not on the *word* Safari). A Finder window or tab will open up with Safari highlighted.
Leaving Safari highlighted, next hold down Command + tap the "I" key.
At the top of the Info window, under General, you'll see "Where." If you see your Safari being in any location OTHER THAN in "/Applications," then your Safari's in the wrong place.
⚠ MOVE IT TO THE APPLICATIONS FOLDER IF IT'S NOT THERE ALREADY.
LOTS of apps, mostly all Apple apps, misbehave or just plain won't work if they're not located at the root level of the Applications folder (like Mail for instance, esp after a major Security Update or OS upgrade. Been there, done dat about a year ago, missing a patch of hair because of dat).
So if your Safari isn't located at the root level of the Applications folder, move it there.
And to be on the safe side after moving Safari, if you had to do that, reboot.
Your Safari ought to work just fine after that.
However, If it's still not behavin' after rebooting, then I'd suggest closing down everything & then "overinstall" Mavericks onto itself (i.e., a 2nd install, right on top of the 1st one). BUT before doing that, make sure that ALL of your Apple apps & Apple utils are located where they are supposed to be, where the Apple software engineers put them I mean. If you're not sure, then dumping them all from the various locations you might have them, & putting them all into the Applications folder at the root level is a safe bet before overinstalling Mavs onto itself.
The latest Safari for Mav, btw, is v7.0.3, which can be downloaded from here:
https://www.macupdate.com/app/mac/15675/apple-safari
You might also check out your MB, MBP or whathaveyou's firmware version. Mav runs best on hardware with fully updated firmware.
KK