When I run into this situation it is usually from a lack of context. The player doesn't feel his character has any reason to be in the situation they are in. The issue is more subtle than not liking the adventure's premise. The issue is that the player as his character doesn't feel any connection to the premise.
The way to overcome this is to give the player a reason for his character to care other than the obvious motivation of adventure, treasure, have been ordered too, etc. The way to do that is to give each character the context in which they exist within the setting. By working with the player and selecting background elements you give them natural motivations for going adventuring.
Think about the setting in which your campaign takes place. Come up with some ideas on the cultures (including those of different races), organizations, religions (the culture that surrounds the worship of a deity).
Then sit down with each players, including the "problem" player, and work with them on creating a background using the above as the context. It doesn't need to be long, a page of notes will be enough. Start this off with the player expressing in general terms what their characters wants to be and do. Then present alternatives from your setting that players can choose. Go back and forth exploring alternatives until both of you are satisfied that you have an interesting background. While you do this for each player individual as a whole manipulate the process so that there are natural reasons for them to be together.
For your problem player, this will provide not only compelling reasons to go on adventures but also natural allies for him to rely on and enemies for you to craft adventures around. Perhaps he associated with the Temple of Light and knows that if push comes to shove he can get a good rate on healing or even raise death.
Also by providing connections between the different players you have the players doing some of the work for you in keeping the problem players going in the adventures. In tabletop and live-action roleplaying I know several normally cautious and timid player do amazing things because a friend or shared goal was involved.