Taking the rope down could create bigger problems, depending on the situation.
If you are alone, you must (1) detach one end, (2) cross the stream, (3) detach the other end, (4) retrieve the rope, and (5) carry it out. While the rope is flapping in the stream during step (2), the rope could get snagged on rocks and vegetation, or you could get snagged in the rope, creating a much bigger problem. And carrying the rope out could impair your safety, from the extra weight, or from snagging hazards in future stream crossings.
If the stream flow is low, the stream is narrow, the snagging hazards are minimal, and carrying the rope doesn't otherwise impair your safety – go for it.
In any case, you shouldn't trust the rope to hold your weight if you didn't rig it yourself, so cross downstream of the rope, preferably without touching it at all.
Ropes and swift water are a dangerous combination, especially in untrained hands. Avoid if possible.
I have an unreliable sense of balance, so log bridges are usually my last choice. You must judge the consequences of falling off any log bridge – How far is the drop? How deep is the water? What would you drop onto? How fast is the current moving? What's downstream? – before you proceed.