There were two main qualifications, one is primarily cultural, and one is really universal.
A host family (or person) would need to be hospitable. Abraham, Lot, and others throughout the Old Testament were "lovers of strangers" (to use an anachronistic expression derived from the Greek word for hospitality). In the ANE, hospitality and being a good host were more highly valued than they are today, at least in the minority world. Today we may say "Make yourself at home," but we don't really mean it. In Jesus' day, a servant would wash a guest's feet, and the host(s) would see to it that their guests were fed well and made to feel comfortable. They would be invited to stay the night and be sent on their way the next day with plenty of snacks for their journey (see Genesis chapter 18:1-8; 19:1-8, especially the phrase ". . . they have come under the shelter of my roof"; in other words, they are my guests, and I am responsible for them, even for their safety).
A host family would need to be amenable, amenable to the teaching of Jesus. Their openness may have been instigated by curiosity or by a sincere desire to learn more deeply about what Jesus and His message were all about. As a contrast, take the townsfolk in Luke 8, who had witnessed Jesus' healing of a demon-possessed man:
"Then all the people of the region of the Gerasenes asked Jesus to leave them, because they were overcome with fear. So he got into the boat and left" (v.37).
Unlike the people in the region of the Gerasenes who in effect said to Jesus, "Here's your hat. What's the hurry?", the "worthy" families who welcomed Jesus' disciples into their homes evidently wanted to hear more about this unusual prophet and teacher whom they may have already heard in person or had heard about through "word of mouth" advertising.
"Unworthy" homes, on the other hand, would be neither hospitable nor amenable to Jesus' disciples or Jesus' message. These hosts might invite the disciples in, but upon finding out what Jesus was really about and consequently taking umbrage at His message, they quickly ceased being amenable! As with homes as well as with towns, if the disciples' reception was less than cordial, Jesus instructed them to wipe the dust off their sandals as they left either the house or the city (or town or village). This act, practiced by Jews in Jesus' day who upon returning home from a journey through pagan lands, was symbolic of (to mix metaphors) washing their hands of those Gentile dogs.
Interestingly, Jesus in His Sermon on the Mount instructed His followers:
"'Do not give what is holy to dogs, and do not throw your pearls before swine, or they [the swine] will trample them under their feet, and [the dogs will] turn and tear you to pieces'" (Matthew 7:6).
The message of the Kingdom of Heaven is the "pearl of great price" (Matthew 13:46). To "throw it" before people who despise its value is, well, unwise, not to mention a waste of time, talent, blood, sweat, and tears. In other words, with an "unworthy" audience, Jesus is telling His disciples--whether in the first century AD or the twenty-first century AD--they are best to move on.
The beloved Apostle John in his third letter commends Gaius for acting faithfully in the way he treated itinerant teachers, evangelists, and church planters, whom John called "strangers" (v.5). Gaius was what you'd call a settler. These travelling servants of Christ, on the other hand, were pioneers. (Thanks to Chuck Swindoll for these two terms!). Gaius and the church of which he was a part supported these travelling preachers financially and by opening their homes to them until their work there was done.
John's expression "send them on their way" meant, in part, that the local church provided these servants of the Name with travelling expenses and perhaps even clothing, food items, other practical items, and perhaps even a love offering for the next church on their itinerary!
Diotrophes, on the other hand, put the kibosh on helping these travelling missionaries, and he even kicked out of the church those who wanted to show hospitality to these strangers. (Remember, a hospitable person is a "lover of strangers.") John had some strong words of rebuke for Diotrophes, of whom he says "loves to be first among [the travelling missionaries]" (vv.8,9).
All this to underscore the importance of hospitality in the life of any local church. As the writer to the Hebrews said,
"Be not forgetful to entertain strangers: for thereby some have entertained angels unawares" (Hebrews 13:2 KJV).