Check a string to see if it is a palindrome
I would prefer the first one for a few reasons. 


It's more readable and easily understandable in my opinion. 
There's only one string, so less overhead. (Which is negligible unless this is being called a lot.)
There's only one array. Reverse creates a second array.


That being said, an actual review is in order. 


Methods that return a boolean value should have names of the form IsSomething or HasSomething. A great name would be IsPalindrome. 
Neither method accepts anything but a string. Numbers can be palindromic too. Wouldn't it be a nice addition if we could pass an integer to it as well? (Without explicitly calling .ToString() before passing it in that is.)
I like that you're only checking for i &lt; j instead of i &lt;= j. It saves a useless iteration when there are an odd number of characters. 
I also like the while loop. It's a lot cleaner than the for loop suggested by another answer. 
I won't speak much on the second method, because I think @elios provided a nice implementation of it.