People will always be more willing to help when they feel like they owe it to you. This classic persuasion method requires a little pre-work, but can make it much easier for you to convince others to lend you a hand when you need it.
The goal is reciprocity, and that requires you helping them out too. Robert Cialdini, the author of Influence: The Psychology of Persuasion, suggests that helping others to advance their goals first is the key. The trick, however, lies in how you go about it. He explains to Harvard Business Review:
Get in the habit of helping people out, and -- this part's really important -- don't wave it away when people thank you. Don't say, "Oh, no big deal." We're given serious persuasive power immediately after someone thanks us. So say something like "Of course; it's what partners do for each other" -- label what happened an act of partnership.
Don't wait to be asked to help with something, just help. Then make it count by explaining why you're willing to help. Tell them, "that's what friends do for each other," or "anything for a team member." The next time you need help from them, you'll probably get it.