The 1060 is plenty capable of running nvenc. All the cards of a given generation all have the same nvenc encoder ASIC (basically, the standalone bit of silicon dedicated for encoding video), so you have the same capability for encoding as any other 10-series card, from 1050 to 1080ti. Running that encoder has no performance hit on the rest of the graphics rendering it does, however it does require extra bandwidth in the form of data sent over the PCIe bus, as well as from VRAM to send data to the encoder's frame buffer. This can be minimized by using the Nvenc (new), which tries to prevent round-trips across the PCIe bus as much as possible, but there will still always be some kind of performance hit if bandwidth is a limiting factor.
That said, nvenc is generally the least system-intensive encoding option, second being QuickSync, all thanks to that dedicated silicon. As long as you make sure to not use the options I listed above, it should be the least impact to gaming performance.