It is unfortunate that many Christian men still use that verse to justify ruling over women. They are able to convince themselves of that, because the word translated "rule" is the Hebrew word "mashal". the creators of Strong's Concordance define "mashal" as follows:

a primitive root; to rule: (have, make to have) dominion, governor, reign (bear, cause to, have) rule, have power.

If the creators of Strong's can be believed, the verse as it was written in Hebrew by the author was not saying the man would merely have responsibility for the woman. It was saying the man would have dominion over her. If the author (Hebrew copier/editor) meant something softer than that, they would have chosen a different word.