Rome did actually attempt to conquer the east coast of the Red Sea and southern Arabia during the rule of Emperor Augustus in the first century BC.

The territory roughly corresponding to modern day-Yemen was then occupied by the Sabaean Kingdom, one of the main incense-producing regions of the ancient world. It also possessed gold reserves and a seaboard facing the Indian Ocean, making it a potentially valuable addition to the empire. The Romans recognised this value and also sought to bring the entire Red Sea under their jurisdiction by conquering both coasts.

In 25 BC a Roman army invaded Arabia and was militarily successful against its opponents but made slow progress towards the Sabaean capital, largely due to the climate and geography of the region. The Roman advance eventually halted altogether when soldiers began to suffer from a scurvy-induced paralysis in their limbs, likely brought on by a lack of vitamin C in the campaign diet. Fearing the "unknown" sickness killing his troops, the Roman commander withdrew his forces from Arabia.

However, the Roman assault nevertheless destabilised the Sabaean Kingdom and caused it to fragment. The new regimes that seized the territory understood the threat posed by Rome and regularly sent embassies, envoys etc. to discourage invasion and promote friendship and trade ties, hence why no further attempt was made by Rome to conquer Arabia.

Source: The Roman Empire and the Indian Ocean by Raoul McLaughlin, a very in-depth and interesting read that I highly recommend if this sort of thing interests you.