Well, not necessarily "obsessively." But it might mean that the first meaning of a phrase that comes to mind is the one that has a sexual connotation.
Take a recent post about a girl with a large dowry. She was well endowed. Someone with their mind in the gutter might first think of the meaning of "well endowed" that means she had large breasts.
Someone whose mind is seriously in the gutter won't be able to hear even the most innocent phrase like "magic wand" without making a joke that makes the phrase mean "penis." (By the way, someone recently pointed out how funny passages of Harry Potter become when you change "wand" to "willy." You have to have your mind at least a little bit in the gutter to laugh at the idea. I did.
Sometimes people like that are very funny. Other times they are just immature and annoying.