Under US law, the use of the converter is irrelevant. The legal situation would be the same if they were posted in MP3 format, or downloaded and played in whatever format they are posted in. The point is making and distributing copies without permission.
The first question is: Is the music protected by copyright at all? If the work is old enough, there is no copyright on the composition. For example, most works of classical music will be long out of copyright. However, the recording itself can be copyrighted, even if the composition is not. In general, if recording was published before 1972, it will not be protected by US copyright. There are some other edge cases where the the recording will not be protected. See This chart for details.
Assuming the recording is protected, the second question is: is the posting legal? That is, was the music posted by or with the permission of the copyright holder? If not, any download or further use would be copyright infringement, although holders are unlikely to sue individuals who download for personal use only.
If the posting of a protected work was legal, the key question is, did the user have permission or some other legal basis. It is generally considered that when music (or other content) is posted to the net, there is an implied permission to download it for personal use. Alternatively and to the same effect, this might be considered in US law to be a case of fair use. But this will stop at personal use.
Any making of additional copies, redistribution of such copies, or public performance of the music will require permission from the copyright holder. In the absence of such permission, it will be infringement, and the holder could sue. Permission may be granted directly, by contract, or by a permissive license. But permission in some form is required for such use to be lawful.