Mundungus might have expected that he'd be doing jobs fairly low in danger.
Mundungus isn't brave, nor did he ever try to claim that he was. When the Order planned to turn some of its members into Harry to transport him, he openly objected to being turned into Harry and the danger that would bring him. He made no effort to hide that, even though he was a member of the Order, he had no desire to risk his life for Harry or the others.
“Everyone here’s over-age, Potter, and they’re all prepared to take the risk.’ Mundungus shrugged and grimaced; the magical eye swerved sideways to glare at him out of the side of Moody’s head.”
- Harry Potter and the Deathly Hallows, Chapter 4 (The Seven Potters)
Later, he explained that the reason he Disapparated was because he panicked.
“I panicked, OK? I never wanted to come along, no offence, mate, but I never volunteered to die for you, an’ that was bleedin’ You-Know-Who come flying at me, anyone woulda got outta there, I said all along I didn’t wanna do it –’ ‘For your information, none of the rest of us Disapparated,’ said Hermione. ‘Well, you’re a bunch of bleedin’ ’eroes, then, aren’t you, but I never pretended I was up for killing meself –”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
We know the Order let him in because he's supposedly loyal to Dumbledore, and having a crook around can be useful.
“How come he’s in the Order?’ Harry said, very quietly.
‘He’s useful,’ Sirius muttered. ‘Knows all the crooks – well, he would, seeing as he’s one himself. But he’s also very loyal to Dumbledore, who helped him out of a tight spot once. It pays to have someone like Dung around, he hears things we don’t. But Molly thinks inviting him to stay for dinner is going too far. She hasn’t forgiven him for slipping off duty when he was supposed to be tailing you.”
- Harry Potter and the Order of the Phoenix, Chapter 5 (The Order of the Phoenix)
He may have chosen to join the Order for the protection being favored by them would offer.
Dumbledore was willing to overlook his smaller crimes, like stealing from 12 Grimmauld Place, likely because he was a member of the Order. After Dumbledore was dead, he might have considered the Order of the Phoenix to be the stronger side, or that he might get benefits from remaining a member of the Order. Money is only one possible benefit. Being able to count powerful wizards like Dumbledore among your allies is an invaluable asset, and Mundungus likely considered that when deciding to join the Order.
It may be similar to Pettigrew joining the Death Eaters without real loyalty to the Dark Lord.
Peter Pettigrew became a Death Eater, but not out of support for the cause or respect for the Dark Lord, but because he thought that being on the side of the Dark Lord would be useful.
“When did I ever sneak around people who were stronger and more powerful than myself? But you, Peter – I’ll never understand why I didn’t see you were the spy from the start. You always liked big friends who’d look after you, didn’t you? It used to be us … me and Remus … and James …”
- Harry Potter and the Prisoner of Azkaban, Chapter 19 (The Servant of Lord Voldemort)
The Dark Lord himself knew this, but kept Pettigrew around anyway because he had his uses.
“My Lord! I – I have no wish to leave you, none at all –’
‘Do not lie to me!’ hissed the second voice. ‘I can always tell, Wormtail! You are regretting that you ever returned to me. I revolt you. I see you flinch when you look at me, feel you shudder when you touch me …’
‘No! My devotion to your Lordship –’
‘Your devotion is nothing more than cowardice. You would not be here if you had anywhere else to go. How am I to survive without you, when I need feeding every few hours? Who is to milk Nagini?”
- Harry Potter and the Goblet of Fire, Chapter 1 (The Riddle House)
In the same way, Mundungus could have had cowardly motivations for joining the Order. Sirius says that Mundungus is very loyal to Dumbledore because Dumbledore helped him out, but even so, his motivations don't have to be entirely noble for them to trust that he wouldn't betray Dumbledore. It's possible, though nowhere is this confirmed, that Dumbledore helped him get out of trouble in exchange for him becoming a member and helping the Order.