https://www.quora.com/Why-do-some-women-object-to-having-sex-in-the-doggy-style-position

Many do enjoy sex in the doggy-style position, but those who "object" may not like it because they find the position:

Degrading - Because of it's association with dogs and other animals who mate in this position, a woman may feel she is being "treated like an animal." Doggystyle is degrading to women so they won't do that position

Submissive - Some women feel that the position is extremely submissive for her, and extremely dominant for her partner, and these women may prefer positions that evoke more equality.

Not Intimate - Because the position does not allow partners to easily kiss or make eye contact, some women may find that the position diminishes emotional connection.

Objectifying - From a political perspective, if a woman finds the position "degrading, submissive and not intimate," she may feel that she is being treated like an object rather than a human being.

Unflattering - Since the position requires her to lift her buttocks and thighs high up behind her, and let her breasts and stomach dangle, a women who is self-conscious about the shape of her body may feel the position makes her look unattractive.

Embarrassing - Some sexual acts simply embarrass people, and this position may make a woman feel exposed or humiliated.

Uncomfortable - Some women may not like the roughness, deep penetration, and lack of clitoral stimulation in this position.

Taboo - Because of religious views or social conventions, a woman may have moral objections to this position.

Powerless - Because the position allows her partner to control the rhythm and intensity of thrusting, allows her partner to use his hands to touch her without her being able to easily touch him, and allows him to see her while she is facing away, she may feel the position requires her to give up too much control.

Too Intimate - In contrast to the lack of intimacy issue above, some woman may find that the position requires a level of trust and connection that she doesn't yet feel with a particular partner.