A safe place to collect the horns is the Fremennik Province east of Rellekka, north of Swaying tree at the rare tree icon. With fairy ring access this spot is not far from a bank, and there is very little competition. There's 6 unicorns in total, but only 2 of these unicorns drop unicorn horns. (The other 4 are black unicorn foals.)
Buying unicorn horns from other players or on the Grand Exchange. This can be somewhat expensive, as the horns are often in high demand.
You can also kill black unicorns in the Wilderness, this is dangerous and it isn't suggested, but there are a decent amount of black unicorns there which always drop unicorn horns.
Creating and then killing Unicows in the Creature Creation activity. You will need a unicorn horn and a cowhide. This is fastest means of getting unicorn horns without purchasing them, as each unicow always drops several horns. This method requires completion of the Tower of Life quest. Wearing any version of the Ardougne cloak from the Ardougne Tasks will cause some of the unicorn horns to be noted when dropped, increasing the number that can be gathered per trip. As noted horns cannot be used to summon unicows, players must ensure that when their final unnoted horn is used, the cape is removed before killing the summoned unicow.