There is always the option that nobody patrols the borders, too. I mean, nobody ever patrolled the sea borders of the British Empire, either. It would simply have been impossible to do so. Rather, the Empire had concentrations of naval power, including home ports, replenishment stations, and then starbase-like strongholds at distant locations, such as China Station. There was little in the way of early warning, and everything worked on the basis of random inspections, random escort, and decisive retaliation if somebody dared.
Yet an enemy fleet could always sneak to within a few nautical miles of British mainland in decisive strength without being noticed - as long as there was nothing faster than a ship of the line to carry a warning... Wireless eventually changed that. But it didn't result in the Empire starting a border patrol.
Starfleet doesn't have much in the way of early warning, either: only the Dominion can monitor sector upon sector with something analogous to radar. Only isolated special areas such as the RNZ are an exception, and those, too, leak like a sieve. Tellingly, we never hear of Starfleet mounting a border patrol. Instead, we get local patrol, protection of point targets such as star systems - the biggest thing we hear of being patrolled is a "space sector" in TOS "The Menagerie", but that's a one-off!
Timo Saloniemi