Yes must steal enough blood to protect his family? sounds better and yes "caretaker" or "loyal caretaker" also sounds better.
But I still feel like just steeling blood whether to feed his family or himself is not compelling enough of an action to sustain my interest in this instance. The fantasy-Goth aspect of vampires aside the story is about a guy struggling to provide for his family like Will Smith in The Pursuit of Happiness. Will Smith's character is a struggling all round good guy underdog, however your MC is a natural killer, all be it rejecting his nature but a natural killer none the less and an outlaw so in a way he's an anti-hero.
I think that an anti-hero struggling to provide for his family is less interesting than an anti-hero fighting a bona fide bad guy. Mostly because I can't see the tangible goal to end the problem he has, that is "to provide for his family" or the equivalent in vampire terms "... steal enough blood to prevent his family?s death."
How long have they been starving for? How long can they go without blood? How many of them are there? If he gets the blood he needs now how will he feed them in a months time?
I don't think these questions should be answered in the logline but are inherently present after reading the current draft of it.
I would suggest shifting the focus of the logline away from the specific form in which he provides for his family and onto an easily understood imminent threat. Perhaps in his pacifism he befriended a person that works for the blood bank who sneaks out bags of blood for him and his brother/family.
Then after his master is killed their lives and source of food are threatened and he must fight to save them all.
I like fantasy and horror films and think this is a nice and refreshing take on Vampire stories (un like the horrid treatment the genre got in Daybreakers and other poorly structured films) so definitely worth perusing.