TL;DR - First try to identify if the wire is supposed to carry electricity (insulators on posts/wire goes to an electrical box/posted warning signs/ect...). There's really not much you can do to check the wire for current without a device aside from listening to it, touching it (potentially dangerous), or throwing something onto it that might react to a current (something wet/metallic/living/etc...). Maybe hold your arm close to see if your hairs stand up? If you have a compass, you may be able to use it to detect a magnetic field around the wire depending on the type and flow of electricity. These methods will only confirm that there is electricity in the wire, but without proper equipment, you can't really confirm a wire isn't charged, and you're operating under assumptions and risk shock by touching the wire.
If you're asking how to identify an electric fence, the answer is that there will be insulators on the posts. Plain wire fences are just stapled or tied to the posts. Also, if the fence emits a powerful enough charge to hurt you, there will be caution, warning, or danger signs posted.
Insulator on steel post
Insulator on wood post & electric gate with insulated handle.
I've climbed over electric fences plenty of times, they use them out here to deter bears from getting into certain areas (like the local dump) and keep deer out of crop fields. First few times I climbed through one I felt nothing, so I thought it was turned off (I was a kid at the time). It wasn't until I was holding two wires wide apart so that my dog could get through and touched the back of my hand to the wire above it that I got shocked.
The thing about being a human is that you're typically insulated against the ground to some degree because of the rubber soles of your boots. Electric fences only really work if you're grounded or touching more than one wire. Even if you do get shocked, it's meant to smart, not injure, at the very most you might get a little red burn. Electric fences don't use a constant current, they pulse, sending out shocks that don't feel too dissimilar to a shock from a BBQ sparker.
Of course this is only true for fences designed to keep animals at bay, if you encounter an electric fence designed to keep humans out, they will typically have big signs indicating that they are charged and will hurt you. Razor wire at the top is another give-away.
EDIT: In response to edit by OP. If you happen upon a cable that you might want to use for a crossing, first of all, use common sense. Ask yourself, "Why is this cable here?" Is it meant to carry electricity, or does it have some other purpose? Your number one priority is still to identify if it is an electrical cable or not. In my part of the world it's common to happen across a forestry cable car for crossing creeks or canyons:
But they're typically a dead giveaway, the cables don't extend beyond the anchors, and there's a basket with pulley wheels.
You are not going to be able to easily access high-power electrical cables, they're put high up on poles for a reason, and won't necessarily carry your weight. The biggest giveaway that a cable is carrying a lot of electricity is that it will hum. Every power cable I've been under in the backcountry is easy to hear especially if it's raining or snowing.
To be brief: there's not much you can do to safely detect electrical current in a wire without some sort of a device, but something that you may have on your person if you're in the woods, is a COMPASS which you could use to check for an electrical current. Hold the compass near the wire, the magnetic field will move the needle if there is a current in the wire.