Yes, it still applies, but it has to be properly understood. It is not a question of worth or independence, but rather a question of role. God looked at Man and saw that Man was not ideally suited to life by himself and that he should have someone to be with. Woman was created to fill this role as a partner and helper to man just as man is a helper to his wife. This passage specifically does not lay out any hierarchy, but rather just the relationship of men and women as complementary.
Later, in Ephesians 5, a more specific relationship is laid out by the instruction for wives to submit to their husbands, but this is easily abused if taken out of context if the follow up that men are to love their wives as their own bodies. In short, the meaning of Ephesians 5 is that if living rightly, it is the husband's responsibility to make choices such that his wife benefit and that in the event of a conflict, wives should trust their husband to put them first.
This is a conflict resolution strategy more than anything else and puts more burden and responsibility on the husband, who is expected to sacrifice for his wife just as Christ sacrifices for the Church. Again, it is not intended to put women in a position of lesser importance or rights than men, but rather to instruct men that they should treat their wives as themselves, which includes not requesting submission unless they truly believe it is in their wife's best interest. Similarly, while men are asked to sacrifice their interest for their wives interest, wives are asked to sacrifice for their husband in following his leadership in the case where he feels that something is genuinely for his wife's best interest.
Ephesians 5 particularly is a hard passage to internalize, but with proper understanding that it in no way establishes one as more worthy or more powerful than the other. It is not an establishment of authority of one over the other, but rather a template of the type of sacrifices that should be made in a healthy relationship. This is a mirror to Christ's relationship to the Church where we submit ourselves willingly to God's will even when it isn't what we think is best and how God does what is best for us.