That's incorrect! Routine use of a path over your property over a large amount of time is a classic case where a public easement is created without the owner's consent. See "Prescriptive Easements"[1]. You absolutely will be required to re-open the path.
There was a post on /r/legaladvice where someone's neighbor sold a plot of land, causing them to effectively be landlocked, but there was a road through OP's land connected to the seller's property with a locked gate to which the seller was demanding access. Basically everyone was telling OP not to allow the seller access, even once, as that would set a precedent.
[1] http://www.nolo.com/legal-encyclopedia/easements-overview.ht...