You need a component for parsing MIME format described by RFC 822/2045 and extensions. .NET framework doesn't include built-in classes for that.
I've good experience with Mime4Net component (it is based on port from Apache mime4j):
Stream mimeMsgStream;
var m = new MimeMessage(mimeMsgStream);
MimeMessage provides DOM for MIME structure and attachment content could be easily extracted. Also note that Mime4Net is free only for non-commercial usage.