Not enough info to go on, the success rate depends on how good your party members are at the task. I'm not quite sure which part you're getting stuck on so: How quests work as far as I know:
Step 1: You set the amount of gold you offer your adventurers to go on the quest for you, it's split among the members, max members available for that is 4. I don't think there are any downside to having max members other than the increased cost, so you pretty much always wanna max out the party. If you don't offer enough payment to them, they won't want to go on the quest.
Step 2: You hire units that have the skills the quest requires (Look for the numbers show in the bottom left, higher the better). It sounds like you're maybe having issues with this part to me. Probably try to wait until you have at least 2-3 of the members with +2 or better in whatever stat the mission requires. I think some of the skills on the right can help too, but I don't know much about those.
GEORGEM said above that higher ethics/loyalty scores can prevent them from running away, but I haven't tested it myself. Makes sense though. That means you also probably want to avoid any adventurers with negatives in those stats (Or just negative stats in general).
Step 3: You wait a day or whatever and they finish the quest or don't.
Since the quest is 1 star difficulty, you don't have to worry about your adventures being underleveled. With the 2 star and up quests, H-G rank might not have enough stats to be successful. Adventurers often level up after a successful mission. So you wanna make sure to try to level up your adventurers before the higher difficulty stuff. For as far as I played, most 1-2 star can be done with starter units, but I failed the 3 star quests a ton, those are a bitch.