This is one of those fights were you might think it's a stomp given Danny's abilities but this is and interesting match up as I'll explain below and the end result is actually surprising.
Danny's Advantages/Win Conditions
So Danny scales to a higher value and has a decent scaling chain, his ghost physiology gives him regeneration she can't overcome by physical means, and intangibility will allow him to phase out of her telekinesis grip or past her forcefields. Possession or him hitting her with ectoplasmic energy is a win though I'm unsure if Danny would actively try to destroy the mind/soul of a dog (though ectoplasmic energy naturally effects the body on a molecular level and those so unless he changed the function of his attacks which he only did a few times it's going to happen regardless).
Brainy's Advantages/Win Conditions
While Brainy is weaker, her telekinesis will allow her to surprise Danny (as he wouldn't know when she'd do it) and he doesn't have the lifting strength to overpower her (plus he doesn't weigh much) so she can throw him around or stop him with ease and his only escape would be intangibility. Her smelling, hearing and mind reading should allow her to not get surprised by invisibility, and adding to her mind reading, this would give her the low down on the type of person Danny is, his powers, his overconfidence, and the container he always carries with him, the Fenton thermos.
My Vote
I'm going to vote Brainy and I'll explain why below. This is a fight that actually gets more interesting when you look at not only the powers but the characters using them.
Yes, Danny has multiple win conditions but they all focus on landing a hit. This is problematic because Danny focuses on close ranged and ranged combat but he still moves well within the Brainy's range even when using his ranged attacks and I'm not joking, despite having decent range, in combat Danny still gets in close when blasting his opponents and this fight starts at 20 Meters. Now while Danny can use intangibility to break the hold, he'd need to keep it active in order to avoid being grabbed again and while it's one of his more used abilities, Danny doesn't hold his intangibility for long periods of time as he tries to get back into the fight. It's true he can also use the natural state intangibility ghosts have but he doesn't use that in character and in that state he wouldn't be able to touch her either.
Another thing that's important here is the characters intelligence. Unlike Brainy, Danny has an overconfident side due to his young age, and has even admitted (in season one) to himself that he needs to start paying more attention in his battles because it's gotten him caught off guard. So given he's battling a dog, he'll not take her to seriously from the start and will be joking at time because that's who he is. Meanwhile Brainy is more serious than Danny and when applying her mind reading will let her know how dangerous he is and given the range she could use his own Fenton Thermos to trap him as she'll know what it's for when reading his mind.
It's also important to note that Danny's reactive evolution won't trigger because she's not stronger than him and the abilities she has, Danny hasn't shown a resistance too. Here forcefields will also see more use than Danny's since she doesn't have ranged attacks like energy blasts to reflect. Now I will point out that since speed is unequalized, Danny's ghost rays are lightspeed, however he doesn't always fire those. He can fire beams of ectoplasm, heat, plasma or light (the last one being the ghost ray) so Brainy can dodge almost every one of them and again she could read his mind and quickly act before hand.
Basically, the reason Brainy takes the win (at least more than Danny) is because she has a smaller arsenal and thus has a higher chance to go for her win method (which is mind read/telekinesis and then Fenton Thermos), along with her being more serious in the fight as opposed to Danny who wouldn't get full on serious yet and who's overconfidence would play a factor. Danny does have more win conditions but they all function at a close range and thus can be countered to a degree because as I mentioned, Danny doesn't hold his intangibility nore will he be unpredictable to someone like Brainy who again could just read his mind.
