Taking care of Tulasi devi is a great service which will give a devotee enormous benefit in his or her devotional life. It is said that Lord Siva once asked His son what plant could produce krsna-bhakti, and His son said that only Tulasi devi can produce bhakti. Tulasi is so dear to Krishna that if one offers to Krishna one Tulasi manjari, the benefit is equal to offering thousands of every other kind of flower, and according to Caitanya-Caritamrta, if a devotee offers Krishna a Tulasi manjari and Ganga water with love a devotion, Krishna thinks, "How can I ever repay such a devotee?" Srila Prabhupada also mentions Tulasi devi in his purports to Bhagavad-Gita, where he says that when Krishna mentions that he'll accept a leaf offered to Him, He actually means Tulasi and that Krishna will not accept any offering made without Tulasi.
We can see from all this that Tulasi is very special and dear to Krishna. Thus, proper care is sure to result in great devotional benefit. The other side of this is that because Tulasi devi is so dear to Krishna, taking care of her poorly and without the proper standards can be quite detrimental to one's devotional life. So, deciding to care for her is a great responsibility which should not be taken lightly.
The first requirement for taking care of Tulasi devi is cleanliness. One who takes care of Tulasi devi should not eat any meat, fish, eggs, onions, or garlic, nor should there be any of these things present in the house in which she is kept. Additionally, if one lives in a place where people cook meat outdoors on a grill and that smell tends to go to where Tulasi devi would be kept, then that is also not suitable (for her or for you). One should also take a shower every morning, brush one's teeth, etc, and must shower immediately each time after evacuating. Hands and mouth should be washed after eating, and the area where she is kept should be clean.
Ideally there should also not be any pets living in the house such as cats and dogs, as they are very unclean. I know this is a sensitive thing in the western countries because people have been taught to ignore the fact that these animals do not bathe or practice any other form of hygiene, lick their genitals and anus, and then lick their fur and you so that they (the people in western countries) can feel better about themselves despite the fact that the relationships they have with other human beings are shallow and short-lived. At the very least, Tulasi devi should be kept in a room where the pet(s) do not have access and the devotee should bathe and put on clean clothes before going into that room.
Internal cleanliness should also be practiced, which includes measures to keep subtle contaminations away such as no illicit sex, intoxication or gambling, and also measures to purify the consciousness, the first and foremost of which being the chanting of the Holy Names of Krishna:
Hare Krishna Hare Krishna Krishna Krishna Hare Hare Hare Rama Hare Rama Rama Rama Hare Hare
and the regular reading of spiritual literature such as Bhagavad-Gita As It Is, Srimad-Bhagavatam, Caitanya-Caritamrta, etc.
After all of these are met, the most important requirement is that you are willing and able to put a lot of effort into taking care of Tulasi devi. Caring for Tulasi devi means there are no "days off", no "Well, I don't really feel like it right now", just firm dedication. If you're sick, she still must be cared for. If you are traveling somewhere, you have to find someone suitable and knowledgable enough to cover for you. Once you take up the service, you are fully responsible.