I seriously doubt it is possible to keep your sailors as healthy as on the shore with Renaissance technologies. Especially, if they are at sea for months. But it is still possible to prevent them from dying.
1. Malnutrition prevention
As others mentioned, food preserving and canning can provide sailors with vitamins. It is better if preserving done using non-heat or low heat techniques to avoid loss of micronutrients. If preserves contain any liquid it must be consumed and never disposed of. This is especially important for scurvy prevention since vitamin C is water-soluble and tends to leach from foodstuffs into the water.
It is important to prevent oxidisation which is the main reason for the loss of nutrients. So, you need to be able to store food in airtight containers preferably in such a way that contents of each container can be consumed within a day or two.
You also need to pay attention to the price of preserves. Hard bread was very popular (and was the staple of sailors diet) not only because it stores well, but also because it is very cheap to produce. It is essentially flour and water. Sometimes just a sprinkle of salt was added to improve taste. (Although the recipe is quite simple, the preparation was not and could take as long as several months.) Most of the food preservation techniques available at the time involved salt or fermentation. Salt was not cheap. So, it was not economically viable to have large quantities of high-quality preserves on a ship.
In order to prevent common diseases and treat them, you have to have doctors and some pre-established knowledge of those diseases. Some form of a primitive germ theory would be a great help. It would allow establishing simple but effective disease prevention procedures such as boiling water and quarantine.
Ship cleaning is essential for keeping the crew healthy. It is also a perfect excuse to keep them occupied. It is better if sailors understand that a clean ship equals better health.
Living in close quarters (and there is no way around it on a ship) on its own creates health risks. Physical risks such as the rapid spread of infections are obvious. However, psychological health is even at a greater risk. So, any ship would greatly benefit from having a priest or a morale officer. Some form of recreation (including sparring or any other physical activity) would reduce occurrences of brawls by providing a way to react on aggressive impulses without resorting to a mutiny.
Modern Western sensibilities and daily showers have no place in a Renaissance setting. Neither they are practical nor particularly healthy. Frequent showers strip moisture from the skin and by drying it make it more susceptible to microbial infections. It is also not possible to carry enough fresh water for showers. However, one of the ways to keep reasonably clean is to use baking soda, vinegar, and wet rags.
For those wondering about practical uses of baking soda and vinegar
Sodium bicarbonate (baking soda) is a mild disinfectant. A mix of water and soda (usually as a thick paste) can be used to disinfect various surfaces. Baking soda also has some fungicidal properties. If mixed with oil or water baking soda can work as antibacterial soap (but do not expect it to kill 99.9% of all known germs). An additional benefit of such a soap is the elimination of body odours.
Sodium bicarbonate has many other useful applications: 1) cooking (as leavening, aka raising, agent); 2) teeth cleaning; 3) laundry detergent; 4) rust removal; and even heartburn treatment.
As @RonJohn mentioned in comments sodium bicarbonate was not available in pure form until the 19th century. If it is the case in your world, you can replace it with natron. It's a naturally occurring mixture of various salts available almost on all continents. I am not sure it will be edible, but it will still work as a great cleaning agent (as proved by Ancient Egyptians).
Vinegar is a mild acid with antimicrobial properties. Aside from its uses in culinary, vinegar is good general purpose cleaner. It is especially effective against salt build-up (cleaning mirrors :)). When it comes to personal hygiene, vinegar is frequently used as a hair rinse (about one tablespoon for a couple of litres of water).
One thing you should be really concerned about is laundry. Keeping clothes and linens clean is paramount for keeping sailors healthy. Unlike our bodies that have all kinds of protection against infection, clothes and linen have none. Therefore they become breeding grounds for insects and bacteria. It is not practical to have a real wash while at sea. But regular airing and exposure to the sun will work as well. UV light from sun sterilises cloth. Airing and shaking mechanically removes loose particles of dirt and skin.
You also want to keep rats and mice at bay since they carry diseases and spoil food. Keeping rat-hunting animals in combination with daily food stores inspections and cleaning might do the trick.
4. Food Storage
Using food preservation techniques and airtight containers will reduce food spoilage. Rodent control measures will protect dry foods.
Traditionally earthenware jars sealed with wax were used for liquids such as oil, beer, and vinegar. Wooden crates were used for meat, fish, dry foods. Bread and flour were usually stored in sacks made of thick cloth. Water was stored in barrels and was prone to spoilage. Many navies used weak beer and wine as healthier substitutes for water (lower risk of bacterial contamination).
Rainwater was collected regularly. It was used for drinking, cooking, washing, and laundry. Your sailors should do the same. It is the only way to replenish your fresh water supply while at sea. So, some kind of vessels should be within easy reach.