Would that make it invite a lot from scrutiny from the city and power company? An inert wire, they can look the other way. A live wire is a potential fire hazard and source of EM interference for utilities.
The wire is many miles long. You'd probably need some decent voltage to even be able to produce any kind of current in it.
Even then, if the eruv was broken, the sensor would only inform you that it is broken, but not where on the entire island the fault is.
(unless of course you monitor each segment of the wire separately, e.g. using capacitance sensing, with internet-connected controllers that send an alert when the capacitance of their segment changes. That would be the practical but boring solution)
Think of it as a telephone line. They do fine for tens of miles. And you have no signal integrity concerns, merely continuity. (Though being able to launch a TDR pulse down it would be neat, since localizing breaks would become trivial.)
