What does a cursor do?

An executor executes, a parser parses, readers read, etc.

Does a cursor curse? If not, what does it do?

Cursors offer no oaths of their own, minced or otherwise.

They simply run, as @ULTRACOMFY notes.

Etymonline offers the following etymology:

cursor (n.) c. 1300 (as a surname) "a running messenger," from Latin cursor "runner," also "errand-boy," from curs-, past-participle stem of currere "to run" (from PIE root *kers- "to run"). From 1590s as "part of aslide rule or other instrument that slides backward and forward upon another part." The computer screen sense is a 1967 extension of this.

Cursors are commonly said to move, though if you don't mind looking past the screen to the hand that conducts the scene, you can also say that one can position the cursor. For a less pedestrian term, consider the more alliterative cruising cursor.

"Cursor" comes from the latin cursor and means "runner". The more English term would be "Pointer", and a pointer points.

Though a cursor does not curse, a curser certainly does :)

A cursor is a symbol that functions as an indicator or identifying point on a computer screen. The cursor icon marks the point on the screen at which the user may interact with the computer screen...

Curser means someone who curses. It may refer to someone who is invoking a supernatural being in order to punish another person, but most often a curser is someone who uses offensive words in a fit of anger. Curser is also a common misspelling of the word cursor. Grammarist: Cursor vs curser