Originally from an email to the foster coordinator from one of our Foster homes.  It deals with introducing a foster dog into a home with cats.
 

Hi Penny,  First, what behavior constitutes "enamored"?  Are we talking about following the cat around, licking it, playing with it
or trying to mount the cat (not an activity I've ever seen, and certainly not to be recommended!). And what is the cat's response?

My two cats are very different in their behavior with the dogs. Thurman doesn't like a lot of physical contact with the dogs, but he
will share space with them.  He put Teller in his place as a pup and Teller respects him.  Floyd loves attention of any kind, from any
species; he lies down with the dogs and tries to play with their feet and tails.  Jet snarls and he goes away, though he always tries
again.  Floyd enjoys being licked by Teller, he solicits attention by rubbing against Teller's face or rolling around in front of him.
Floyd also tries to groom Teller and he gets in Teller's crate with him at bedtime (we once inadvertently locked them in together!).  As a puppy Teller was just awful to Floyd, grabbing him by body or tail, chasing him, pouncing on him.  Floyd took it all in stride, never bit or clawed Teller.

In my experience, cats usually set the limit on the kind of contact  they will tolerate, and their response to the dog is hard to change.
The cat's response seems to depend on what its previous experience  has been.  All things considered, its much easier to modify the dog's behavior. Also, the goal is not to make these two into bosom buddies,  just to keep everyone safe and reasonably happy.  You don't want either party to bite the other; that only makes things more difficult  in the future.  I don't allow any interspecies chasing behavior because I think that could escalate very quickly to biting or  killing.  Cats chase cats, dogs chase dogs.  Teller is not allowed to join a cat-cat chase.  When he tried as a pup I grabbed him by the collar and pulled him away and scolded him.  A nice solid recall comes in handy too.

For everyday management, I would recommend that the cat and dog be separated when they can't be supervised, for the safety of both. The simplest solution is to crate the dog (which your foster home is probably doing already).  If the cat is taunting the dog in the crate, shut the dog in a room away from the cat.  The litter box should be placed somewhere inaccessible to the dog, but convenient for the cat.  If the cat is bothered by the dog while attempting to use the litter box, the cat may find somewhere else to go.  When the cat and dog are loose together in the house, the cat should always have an escape route  (up onto shelves or counters, under furniture, under the baby gate) and a comfortable area where they can be away from the dog.  I use a baby gate to keep the dogs out of a downstairs room where the litter boxes and favorite outdoor viewing perch are located.  I feed my cats in their crates so that they don't have to fend off dogs and so the dogs can't get to the leftover cat food.

The dog can be trained to leave the cat alone, provided the cat is not provoking the dog.   I have not been able to teach Teller to
leave Floyd alone because Floyd keeps soliciting attention from him.

Try putting a leash on the dog and monitoring his behavior around the cat.  Don't wait until the cat is spitting and clawing and the dog is in full chase mode before intervening.  I don't know if its easier to teach the dog not to touch the cat at all or to allow some
interaction.  I guess I would suggest that you watch the cat's reaction to the dog and let that guide your actions.  You want to
establish limits.  If the dog is sniffing the cat and the cat doesn't seem to mind, then I would just tell the dog, stop, enough, leave it
now.  That behavior was acceptable, but you want to stop it before it becomes something unacceptable.  That way the cat doesn't have to run (provoking a chase) or attack.  If the dog is chasing the cat, or mouthing the cat (and the cat doesn't like it),  pop the leash and tell the dog to "No! Leave it!".  As always, its nice to have a solid recall!  If the dog is being aggressive with the cat, chasing and really biting, then these folks have a real problem which they may only be able to manage by keeping the cat and dog separate (its hard to overcome that prey drive).  The dog may just lose interest in pestering the cat after a while.  After all the dog is new to the house and the cat is an interesting part of his new environment.

You may also find that the cat will use just enough force to establish acceptable boundaries with the dog.  Thurman did this with
Teller.  I came upon Thurman, standing on his hind legs with Teller's upper lip held firmly in his teeth and his paws on Teller's muzzle. Poor puppy Teller was just rolling his eyes, though he outweighed the cat by a good 40 lbs.  He had obviously been pestering the cat who had had enough.  I let them be like that for a few seconds then I gently removed the cat.  Teller has never bothered Thurman since.

I think some of this comes down to alpha and pack leadership too. The dog needs to learn that the cat, like the food on the counter or anything else the dog may want, is MINE, and he can't have it just because he wants it or can reach it.  Once the dog settles in a finds his place in the pack (below all the humans!) it will be easier to modify the dog's behavior toward the cat.

Sorry this is so long, don't know if I've suggested anything you wouldn't have thought of yourself.  The first days or weeks are the
hardest, but everyone eventually settles down.  Cats are pretty smart and they learn to deal with those big dumb dogs.  Sometimes they're even nice to each other, like when Thurman pushes the biscuit tin off the counter and it pops open spilling dog biscuits all over the kitchen floor.

Cathy

ps:  Also, a tired dog doesn't bother cats.  Exercise cures lots of  behavior problems.