Currently, there’s no legal way to sponsor someone so that they come to the US and receive asylum, even considering that there were ways to obtain asylum through sponsorship programs such as the Private Sector Initiative. The reason for requesting asylum also doesn’t matter, whether it is for political, sexual, race, or other reasons.
At the same time, it’s worth mentioning that some refugees can be detained by the US government agencies at the border if the government considers them dangerous to the society. In that case, you can sponsor this refugee and officially declare your determination to support this refugee so that the US detention office releases them. Usually, immigration advocacy organizations support refugees that were detained, so you may contact them to find out whether you can sponsor a refugee.
The US immigration law specifies that those refugees with immediate family members have the right to receive asylum with their family. Note, however, that the immediate family members only include your spouse and unmarried children who are under 21 years of age. Any other family members, such as your parents, brothers, sisters, or cousins, aren’t considered your immediate family. Hence, they can’t be able to receive asylum as derivatives from you.
To make sure that your immediate family members can receive asylum after you, you need to prepare evidence to prove their relation to you.