"These people are obviously reaching the end of their rope," Redmond told a news briefing in Geneva.
"They see no future in Somalia and many of them are so desperate that they are willing to risk their lives and the lives of their families to escape."
The UN agency said it was working "to convince people not to get on those very dangerous smugglers' boats", and is also working to provide aid to those arriving in Yemen to alleviate pressure on the host government.
Somalis are fleeing heavy fighting that worsened two months ago when armed groups stepped up an offensive against the internationally-backed government of Sharif Ahmed, the Somali president.
Mortar attacks by al-Shabab disrupted a parliamentary session on Monday as heavy fighting erupted between the groups and African Union-backed government forces, officials said.
At least seven civilians were killed and 18 others wounded in fighting, police in the capital said.
The parliament was meeting for the first time since al-Shabab and Hizb al-Islam groups launched an anti-government offensive in May.
The violence comes as Somalia's president prepares to meet Hillary Clinton, the US secretary of state, during her seven-nation trip to Africa next week.
The meeting is scheduled to take place on the sidelines of an annual trade forum with sub-Saharan countries, which will be held in Nairobi, Kenya's capital, on August 5, the US state department said on Monday.