Taegan Goddard flags a letter to the editor Rand Paul wrote in 2002 clearly stating his opposition to prohibiting private businesses' discrimination.

In noting his opposition to the Fair Housing Act, which outlawed discrimintion in housing, Paul argued that act "ignores ... the distinction between private and public property."

He continues:

Should it be prohibited for public, taxpayer-financed institutions such as schools to reject someone based on an individual’s beliefs or attributes? Most certainly. Should it be prohibited for private entities such as a church, bed and breakfast or retirement neighborhood that doesn’t want noisy children? Absolutely not.

Decisions concerning private property and associations should in a free society be unhindered. As a consequence, some associations will discriminate.

[...]

A free society will abide unofficial, private discrimination – even when that means allowing hate-filled groups to exclude people based on the color of their skin.