Yes, it was a perfectly common ocurrence to play dice and card games. When at sea, though normally busy, there are moments of absolute dullness with nothing to do, so pastimes were at the order of the day. Even the officers would partake in card games, sometimes a bit too much as my main man Gonzalo Fernández de Oviedo says about Alonso Quintero, a captain he knew:

So, I wanted to say that here, for the matter requires it. And even the best of the pilots who were with the adelantado, Alonso Quintero he was, and I think that none of the three so were; and even if he was, he would not understand, for one things is sailing by stars and another is sailing by routes. I knew him well, and he was a competent sailor, but not a scholar of the quadrant, arbitrary to the routes and common knowledge, more affectionate to the deck of cards than the rest; but ignorant of the astrolabe.

Of course, cards and dice were not the only games, board games like chess were also popular. Bear in mind that at that time, at least in Spain, chess was not exactly a reputable game, it was as disreputable as gambling, as in chess people gambled some serious sums of money. This general lack of reputation is perfectly visible in the amount of municipal ordinances from the time mentioning or banning cards, dice, and gaming boards, such as the 1585 Ordinance of Madrid, article 5: Furthermore, it is commanded that no person dare gamble in this Court any banned games, or cards, or small cards, or have in his house public boards, under penalties contained in the laws and edicts of His Majesty.

Knowing the realities of the sea is a different matter altogether. Hernán Cortés, wise man that he was, when issuing the rules that Álvaro de Saavedra should observe on his voyage opted for some tolerance towards games, as the sailors need pastimes. He is quite literal in this regard:

Item: as the main cause of blaspheming are card games and dice, shall you also prohibit, by proclamation or other means, anyone play those games, and you shall further punish it most rigorously; and because in the armies, specially the sea ones, there is need of some sort of recreation or pastime, you shall allow a moderate quantity of play*, at your consideation, always in your presence, and in the absence shall it be in the presence of the captain, as thus shall blasphemy be avoided, and people will have some exercise*

He is not wrong about blasphemy, and the archives are full of quite hilarious procedures for blasphemy started against people who uttered some sort of expletive after a bad streak. A particular example I love, from the Canary Islands: After having lost four or five games of triumph, losing each game a hen, it was said that he said I renege from God and the Virgin, and if they or any saints are in Heaven, should they have enough power so I don't lose five games in a row

Being far from the main card production centres, people had to get creative in order to get hold of a deck. Bernal Díaz del Castillo mentions that the Spaniards, not having anything better, would use the leather from broken drums to make cards, and he goes as far as to name the maker, a certain Pedro Valenciano. This can be either a family name, or more likely a geographic reference, particularly knowing that Valencia was a relevant city in the printing business and in the card marking one. Lacking leather, carton, or paper, other materials could be used, like leaves of copey, a plant of the Indies, which according to Gonzalo Fernández de Oviedo was perfect for the making of cards, very durable, and very able to hold the ink or paint.

What games were played? An enormous amount are accredited in the Spanish Golden Age, numbering in the several dozen, like the malilla (mentioned in the famous Lazarillo de Tormes), the cinquillo (still popular today in many provinces), the tresillo (still relatively popular throughout Spain), veinte y una, flor, renegado, and a long etc. For the card games, the most authoritative book out there is Léxico del naipe, by Inés Chamorro García.