The primary responsibility rested on the people seeking marriage, either the potential spouses or their parents. A 948 Church synod encouraged nobles to maintain extensive family trees and ancestor-lists for the sake of calculating degrees of relationship, and it seems like a lot of them did. There's sort of a mini-epidemic of 10th-12th century royals seeking to marry Byzantine princesses, because they wanted someone royal but they were too closely related to all the princesses of Western Europe.

At the further-removed degrees, consanguinity tended to come up mostly as an excuse for people who wanted to divorce for other reasons. Louis VII and Eleanor of Aquitaine is probably the most famous case of this today. (The popularity of this practice, in fact, led the Church to restrict the limits back to four degrees in 1215. Oops.) Cases of closer relationship mostly meant third, fourth, maybe fifth degrees. Almost no one married within one or two degrees--keep in mind that 'two degrees' was marrying your sibling.

For closer relatedness, there was first of all the general social disapproval--there are some satirical poems about incestuous royals, for example. And when the Church got involved, things could get impressively heated. Capetian king Robert II had trouble finding a suitable royal but eventually, in 996, married a Burgundian princess--who happened to be the granddaughter of his great-aunt. Two different popes and at least one Church council pronounced anathema on Robert for this marriage!

Lack of record-keeping on the local level would almost certainly have made it impossible to calculate consanguinity out to six and seven degrees for the vast, vast majority of the population. I'm no historian of incest, but I think it's fair to speculate that general social distaste and gossip would have guarded against the closest degrees even among peasants, however.

Sources: Constance Brouchard, "Consanguinity and Noble Marriages in the Tenth and Eleventh Centuries" (Speculum 56, 1981) is probabably the best investigation. James Brundage, Law, Sex and Christian Society in Medieval Europe and R.I. Moore, The First European Revolution, talk a little about how the Church and nobles wielded consanguinity as a political weapon.

banned marriages within seven degrees of consanguinity, meaning that you couldn't marry your sixth cousin,

No, that isn't what it meant because that isn't what was meant by "degrees of consanguinity". There were two ways of counting consanguinity and neither of them meant what OP seems to think it means.

Degrees of consanguinity were numbered by counting the generations to the person both were descended from. Under canon law, you count from both prospective spouses (not counting the actual ancestor), and the longer count is the one that applies but under Roman Civil Law, which is the computation OP refers to, you add both counts together. Under this calculation, six degrees of consanguinity would arise if we had a great grandparent in common. i.e. You were my second cousin. Seven degrees of consanguinity would arise if you were the child of my second cousin.

Of course if we had several ancestors in common the calculations could get more complicated.

Reference: For a detailed discussion of the calculations and of the various limitations that applied at different times and different places see the New Advent Catholic Encyclopedia article - Consanguinity (in Canon Law)

Eleanor of Aquitaine married the King of France, her 3rd cousin. 15 years later, she hated him and wanted out, so used laws about marriage to get the marriage annulled. She then married the future English King Henry II, who was also her 3rd cousin. So it could be ignored or used to one's advantage.