The answer lies below...These answers are presented in order of increasing accuracy, assuming the most literal interpretation.

1. One wrong answer is that there are 2,401 (74)kits, cats, sacks, and wives going to St. Ives. This is inaccurate any way you slice it, as there are actually 2,401 kits, 343 cats, 49 sacks, and 7 wives, which brings me to the next answer:
2. 2800 (74 + 73 + 72 + 7). This is arithmetically correct, but note that these kits, cats, sacks, and wives are not "going to St. Ives".
3. The answer commonly arrived at after thinking about the problem is 1. Just you. "As I was going to St. Ives..." begins the rhyme. This is the generally accepted answer.
4. Yet, solution 3 answers the question "...How many were going to St. Ives?", but that is not the question. It reads, "Kits, cats, sacks, and wives, how many were going to St. Ives?" The answer to that question is zero. There are no kits, cats, sacks, nor wives going to St. Ives. The only thing going to St. Ives is you, and You are not a kit, cat, nor sack. If you are a wife, go back to #3, and then go tell your husband you love him.

In response to cahla, below:

Maybe you should read my reasoning before you scoff at my answer. The riddle asks how many kits, cats, sacks, or wives are going to St. Ives. You are none of those.
If I said to you, "Die Hard movies, how many do you own?" (sure it is a bit odd in the wording, but it is poetic), would you answer with, "Well I own Clueless, so I guess that's one"? No, cause I didn't bloody ask about Clueless, just as the originator of the riddle didn't ask about you.