Here is an extract from a conference paper I wrote last year on the treatment of adultery (among the lay elite, not just emperors/kings):
In the early medieval period the church was struggling to bring polygynous northern Europeans under the institution of marriage. In the late ninth through the tenth centuries the male nobility could change wives with ease and with little reference to the church. Not that lay society ignored all of the church’s moral preaching on marriage. Even though the church lacked real jurisdiction, great lengths were taken to avoid incestuous marriages – even though this greatly restricted the pool of potential pairings [sixth cousins, cf. 1215 Fourth Lateran Council]. By the twelfth century, however, the church had extended its jurisdiction over the dissolution of marriages in a manner generally acceptable to the lay nobility. Yet this was not an act of submission, in fact it could be quite the opposite. The nobility had spotted and now actively exploited a loop-hole that allowed them to return to a similar mode of serial monogamy. By marrying within the boundaries of consanguinity, the nobility realised they had an established ‘get-out-of-marriage-free’ card. Pierre the Chanter records a knight explicitly stating he would marry within the third degree and if his wife displeased him he would be able to have the marriage dissolved. In the long term this played into the church’s hands as repeated sermons on monogamy were internalised while the church retained its jurisdiction over divorce and dissolution – or, perhaps, they could not object when the loophole was closed.
The purpose of the paper was to establish divergence between clerical and 'chivalric' attitudes to marriage (c.1150-c.1220) I concluded that there was a tangible difference but that is rather tangential to your question.
To continue with a comparison of William Marshal and Eleanor of Aquitaine:
The historical figures of William Marshal and Eleanor of Aquitaine offer two very different medieval approaches to adultery. The first comes from the biography commissioned by the Marshal’s son within seven years of his death, in 1219. The L’Histoire is the only documentary source to openly make the accusation of that the Marshal engaged in a sexual affair with the wife of Henry the Young King, his patron and liege lord. While such rumours might have surrounded the Marshal the lack of any corroborative evidence from contemporary historical sources suggests that this was a later fabrication. Not only that but the Marshal was, soon after the supposed accusation, leading the Young King’s tournament teams. Such affairs have been recorded and often with serious consequence, in 1175 there was an infamous case where Philip I, Count of Flanders, discovered one of his mesnie knights in a secret liaison with his wife. The knight, Walter de Fontaines, was beaten by the household butchers and hung head first in a sewer until he suffocated – extra- judiciously, of course. Why then did the biographer, Jean, decide to include this fabrication in a work celebrating the life of the Marshal? The presentation of the accusation is important: it is voiced by the Young King’s courtiers who are jealous of the Marshal’s influence, who believe him to have committed lèse majesté in his arrogant assumption of his own banners and the cries of his men, ‘God is with the Marshal!’ Yet the true reason for their plot is the fact the Marshal is fornicating with Queen Margaret (Mais cel est la verité / Que il fait a la reïne). The author by placing the emphasis on the alleged adultery was able to sidestep justifiable concerns over lèse majesté and have the Marshal compared favourably to the two of greatest romantic lovers of queens: Tristan and Lancelot. The Marshal is reported to have offered to fight any three of his accusers and offered to cut a finger from his sword hand before doing so, thereby emulating the Lancelot-Graal cycle which had begun ten years previously. The inclusion of this likely apocryphal incident, according to David Crouch, leads to a seedy enhancement of the Marshal’s reputation. That the Marshal would be considered capable, or just worthy, of the love and seduction of a queen signals a society that values love as distinct from marriage.
Such a glowing reflection of courtly ideals was not forthcoming in the case of Eleanor of Aquitaine. Her marriage to her cousin Louis VII of France was incestuous, in the fourth and fifth degrees of consanguinity, though this was not apparently known to the two when the marriage took place. Rumours spread quickly and after a scandal in Antioch, during the Second Crusade, the king and his wife were reconciled by Pope Eugene III at Rome, who forbid any further mention of their consanguinity and ordered that they share a bed.vi However, within four years, and one more daughter, the marriage was dissolved. Contemporary clerical commentators hinted at the scandal which had occurred in Antioch and which culminated in Eleanor informing Louis of the illegality of their marriage due to consanguinity. John of Salisbury, who had been present at the Papal curia at the time of the reconciliation and provides the most reliable account, suggests only that Eleanor’s close relationship with her uncle, Prince Raymond of Antioch, was considered suspicious by Louis’s courtiers. He does imply that those suspicions were confirmed when Eleanor requested to remain behind when Louis made preparations to depart. It is after the rejection of this plea that Eleanor mentions the illegality of their marriage. William of Tyre, writing 20-30 years after the fact, is far less charitable: Eleanor is a foolish woman who actively took part in Raymond’s attempts to sabotage her relationship with Louis and who, ‘contrary to her royal dignity, she disregarded her marriage vows and was unfaithful to her husband.’ This representation of a monarch whose actions cause disruption to the stability of society was a form integrated into romance literature, moreover, one of Eleanor’s daughters, Marie Countess of Champagne, commissioned the work Knight of the Cart, one of Chrétien de Troyes's only romances to deal negatively with marriage as an institution.
Now it is actually slightly unfair to compare committing adultery with a queen and a queen committing adultery. But the point is that it could enhance the reputation of a knight that a queen might love him (this is also the period when the Lancelot-Graal cycle begins, c.1220). A queen who loved others was a threat to social stability. But if we look at a corpus of Breton troubadour songs the manner in which adultery on both sides is treated is fascinating.
Edit: sources.
The question is about kings cheating on their wives, rather than queens cheating on their husbands.
One way of assessing how honest they were about this would be to look at the number of married kings or emperors who openly acknowledged children by women other than their wives, as this would be admission of adultery.
Obviously, absence of acknowledged illegitimate children is not proof of absence of adultery, so it's possibly an underestimate of the actual rate. However, the ease with which kings apparently recognised the fruits of adultery suggests they had little to be scared of.
The fact that in England there was an official surname (Fitzroy) conferred on illegitimate offspring of the kings, and that in France the title Maitresse en Titre (official mistress of the king) was a quasi-official court appointment bringing with it a suite of rooms and a household of servants, suggests that the social ramifications were not much to be scared of.
A quick search of Scottish and English kings suggests it was more common for them to admit they had cheated on their wives than not.
Scottish kings:
England (later England and Wales)
As you can see, it was much more normal for the kings to be open about having cheated on their wives than not. I think polygyny is a good word for it, and there seem to have been few social ramifications for the king to fear, although there could be risks to the succession if, for example, the legitimate offspring were female and the illegitimate ones were male, or (as with Monmouth's rebellion) when a king's illegitimate son was popular and his legal heir (in this case, James VII and II, the brother of the previous king) was not, or was arguably disqualified (in James' case by his Catholicism).
It's worth noting that among the 11 kings listed above who acknowledged no illegitimate children (excluding Edward V of England, who died as a child), 3 had no children with their wives either, leaving it possibility that the lack of evidence of adultery may be due to infertility rather than continence. 4 of the 8 kings who had only legitimate children also had a reputation for being unusually devoted to their wives. 2 of these 8 - Henry III and Henry VI - were possible candidates for canonisation. This suggests that not cheating on your wife when you were king may have been seen as something extraordinarily, exceptionally good.
Double adultery, where both partners are married to someone else, may not have been quite so acceptable, as there was an aggrieved husband to contend with (the aggrieved queens are less prominent). Chroniclers of King John's reign specified that he was not just adulterous but that he committed adultery with the wives and daughters of barons. This may not be true - chroniclers had plenty of other reasons to dislike King John and to try to blacken his name - but the salient fact is that it was thought it would blacken his name more than accusations of just plain old cheating on his wife. (see Ralph Turner's book King John: England's Evil King?).
TL:DR social ramifications, what social ramifications? (OK, a little bit of social ramification).