Your clue here is the fact that the NIV version doesn't include the word "never" and in fact contains the invitation to ask for however much dowry they like ("Make the price for the bride... as great as you like").
The King James Version of the Bible is written in archaic language.
The Biblical sense of the idiom "never so" is rarely encountered today and would confuse many native speakers, especially out of context. (Of course, in everyday language, "never so" and "never so much" are often used with a straightforward meaning.)
The Oxford English Dictionary defines the Biblical usage as follows:
In clauses and phrases with concessive force, denoting an unlimited amount.
a. never so: (with an adjective or adverb) as —— as could be.
So "never so much" here means "as much as could be".
The OED includes citations from Dickens, Bram Stoker, and - more recently - Terry Pratchett. Here it is easier to understand than in the Bible, because it seems to have the implication "never so ... as now" or "never before so...".
1844 Dickens Martin Chuzzlewit xlii. 488 [They] worked..to make the door secure; but though they worked never so hard, it was all in vain.
1897 B. Stoker Dracula xxiv. 328 To sail a ship takes time, go she never so quick.
1983 T. Pratchett Colour of Magic 30 There are certain spells that can prevent the life departing from a body, be it never so abused.