From the Super Mario Wiki, the Mario encyclopedia
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The Border Bros. are the Hammer Bros. that guard the Beanbean Borderlands in Mario & Luigi: Superstar Saga. In order to pass into the Beanbean territory, Mario and Luigi must complete their game of Border Jump, which is basically jumping across a rope.
The player must avoid hitting the rope three times with either character or they will have to restart the challenge. After beating them, Mario and Luigi have the option to come back and challenge them again, but the game is harder every time the brothers beat it. Replaying Border Jump earns them more Hee Beans. Before they continue on with their adventure, the older Border Bro. gives them the Beanbean Map.
One of the Border Bros. wears a red flag atop his helmet while the other wears a blue one; although the Border Bros.' names are unknown, it is confirmed that the one clad in the blue-flagged helmet is the younger of them. They are larger than regular Hammer Bros. and more closely resemble Sledge Bros. Due to their shape and the flags atop their heads, they may be a reference to Tweedledum and Tweedledee from Through the Looking-Glass.