How can I set a h.264 profile level with ffmpeg?

When encoding with libx264, you can set the H.264 profile and level with:

• -profile:v – one of high, main, or baseline (and others, but this is irrelevant here)
• -level:v – as defined in Annex A of the H.264 standard, e.g., 4.0.

For example:

ffmpeg -i input.mp4 -c:v libx264 -profile:v high -level:v 4.0 -c:a copy output.mp4

Here we've just copied the audio stream since it won't be affected.

The output will have the correct profile and level set in its metadata. You can check this while encoding, where x264 says something like:

[libx264 @ 0x7fb26103a000] profile High, level 4.0

MediaInfo can also help you analyze container and codec details.

Of course, reencoding the video will degrade its quality to some extent, given that you're applying a lossy conversion again. Try setting the -crf option to influence the constant quality parameter. The default value here is 23, while values between 18 and 28 are considered sane. Lower means better quality. If your input has a bit rate of up to 65,000 kBit/s, chances are it'll still look pretty good after conversion though.

In reference to your comment, try this command:

ffmpeg -i input.mp4 -map 1 -c:v libx264 -profile:v high -level:v 4.0 -c:a copy \
# copies all global metadata from input.mp4 to output.mp4
-map_metadata 0 \
# copies video stream metadata from input.mp4 to output.mp4
-map_metadata:s:v 0:s:v \
# copies audio stream metadata from input.mp4 to output.mp4
-map_metadata:s:a 0:s:a \
output.mp4

Cheers