"guide the course of a vehicle," Old English steran (Mercian), stieran (West Saxon), from Proto-Germanic *steurijanan (cf. Old Norse styra, Old Frisian stiora, Dutch sturen, Old High German stiuren, German steuern "to steer," Gothic stiurjan "to establish, assert"), related to *steuro "a rudder, a steering" (cf. Old English steor "helm, rudder," German Steuer and first element in starboard), from PIE *steu-ro- (cf. Greek stauros "stake, pole"), from root *sta- "to stand" (see stet).
The notion is of a stiff, upright pillar or post used in steering. To steer clear of in the figurative sense of "to avoid completely" is recorded from 1723. Related: Steered; steering. Steering committee in the U.S. political sense is recorded from 1887.
"young ox," Old English steor "bullock," from Proto-Germanic *steuraz (cf. Old Saxon stior, Old Norse stjorr, Swedish tjur, Danish tyr, Middle Dutch, Dutch, German stier, Gothic stiur "bull"), perhaps from PIE *steu-ro-, a root denoting "strength, sturdiness" (see taurus).
So, both the verb and the noun are traced back to the same Proto Indo-European root meaning "strength, sturdiness". The etymology I have posted does not link the two beyond suggesting that they have independently arisen from the same root.
This kind of convergent evolution goes very much against my training as a biologist which has taught me to look for the most parsimonious explanation of an observed lineage. In this case, that would be to have one sense of the term develop from the other. At the time when these words entered English, the only things people could steer (drive) were various types of boats and carts. Carts which were often pulled by oxen, by steers.
I would guess that steering as a verb developed from driving steer drawn carts. Is there any evidence for my suggestion or am I just spinning pretty stories?