In this quotation, the only possible answer is C - "another".
The reason for this is that all of the other options require an article before "cup".
A - Would you like drinking a cup of blood?
B - Would you like having a cup of blood?
D - Would you like tasting a cup of blood?
All of the other options require the article, so the only correct option is "another".
C - Would you like another cup of blood?
As a note, because it came up in the comments, adding "a" to options A, B, and D do not make them mean the same thing as sentence C. They become grammatical by adding "a" but the meaning is quite different.
Sentence C is offering the listener a second cup of blood - "another" implies that they've already consumed at least one cup of blood.
The other three sentences are asking a theoretical question along the lines of
Do you think you would like to [enjoy the action of] drink/have/taste a cup of blood?
This phrasing assumes that the speaker believes the listener has never drunk/had/tasted blood in the past.
Why "a" instead of "the":
"The" is certainly possible but I'd argue that it's less common. Generally, if someone offers you a glass of wine (or blood), they don't offer you "the" (only) glass of wine, they offer you "a" (one of many) glass of wine.
Without some context, it's difficult to know whether there is only a single cup/glass or if there are several and you're being offered a single one.
Let's look at the sentences with "the" and what they mean:
Would you like drinking the cup of blood?
This is certainly possible but, as I mentioned above, it implies that there is only one cup. This also has a speculative quality to it, as if you're asking not, "do you want to drink it" but, "how would you feel about drinking it"?
Personally, as a native speaker, should this be the case, I would prefer the construction:
Would you like to drink the cup of blood?
The same issues occur with the other two examples.
My preferred "the" phrasings of the other two would be:
Would you like to have the cup of blood?
This implies more of a sense of ownership of the cup rather than consuming the blood it contains.
Would you like to taste the cup of blood?
This sounds a bit odd because it sounds more like you're asking if they would like to taste the cup, which happens to be full of blood... which is silly, since you can't eat or drink a (presumably) inedible vessel. So, my truly preferred version of this would be:
Would you like to taste the blood in this cup?
Regardless, the only answer above that works is still "another".