Never, under any circumstances do you put down your weapon in such a situation. Ever.
A person only grabs a hostage out of desperation. The hostage is their last hope. They know this. Without that hostage, they are a dead man. If a person is stranded alone in the desert and they have a full canteen of water, they aren't gonna throw that water away. In 99.9% of the situations, the hostage taker isn't going to kill his victim either. Because to do so gives up his lifeline, his last possible chance to survive. So he may talk big, bad and ugly and make all sorts of threats, but it is highly unlikely he will actually kill the hostage. To do so would be to kill himself.
So the only course of action is to keep your weapon. If you put it down, you are dead and the loved one/friend he has taken hostage will be dead. That is a 100% bad outcome. However, by keeping your weapon, there's a good chance that you live and the hostage lives. And even in the worst case scenarios, some of the good guys are likely to live. So don't put down your weapon! Kill the attacker if the opportunity is available and if you are sure of your gun and your own ability are capable of doing so with minimal danger to the hostage. And if the shot isn't available or if other conditions make it too dangerous to shoot, then use the standoff situation to negotiate with the attacker until he makes a mistake and then nail him. If an opportunity for a shot doesn't become available, then start negotiating a means of escape for the individual in which everyone can walk away, taking into account your own safety and that of the hostage (in other words don't allow him to gain any advantage such as getting to cover while you and the victim are exposed). If the guy grabs a hostage, he is desperate for a means out of the situation. If nothing else is available, chances are he'll be happy to get any deal he can. He will just be worried about surviving the next 5 minutes at that point.
But never, ever put down your gun. It is the worst thing you could ever do and will have the worst outcome of all the possible scenarios.