With MIME, a message and all its attachments are encapsulated in a single multipart message, with base64 encoding to convert binary into 7-bit ASCII - or on modern mail servers running Extended SMTP, optionally full 8-bit support via the 8BITMIME extension.
Prior to the availability of 8BITMIME implementations, mail user agents employed several techniques to cope with the seven-bit limitation, such as binary-to-text encodings (including ones provided by MIME) and UTF-7.