Mr. Henry's personal fortune (significantly under $1 Billion) is probably not what Liverpool fans are looking for - he's no Saudi oil heir or Internet gazillionaire.
He has had success in his ownership of running the Red Sox which went from being a much beloved but ultimately unsuccessful team over the course of 80+ years to winning two championships shortly after he took over.
This is not going to be a Chelsea/Man Citeh approach where Henry/NESV is just going to throw money at every player that moves. The rise of the Red Sox was predicated on innovation in the use of statistical analysis in evaluating players and finding relative bargains based on how other clubs were evaluating players. I won't get into the details because if you don't know about baseball, it will all be a bit too arcane and if you are into baseball, you probably already know the "Money Ball" story and how it moved from Oakland to Boston.
What I see for Liverpool is a similar version of the same approach. Henry will likely be very low visibility and he will likely try to find the Liverpool equivalent of Theo Epstein the "whiz kid" General Manager who is widely credited with building the Red Sox. My sense is that Liverpool will approach success closer to the way Arsenal have under Wenger than any other model I can think of. They are unlikely to compete for the biggest names but they'll develop their own formula for winning and look to buy undervalued players that fit that system. From time to time, they'll compete for a big name that fits their system particularly well but they aren't going to go after every big name player and then try to figure out how to align those players once they're in place (we're looking at you Citeh).
Overall, I think this is a great outcome for