to help, aid, relieve, to ferry, cross
濟 centers on providing relief or assistance to overcome a difficulty: helping, aiding, or relieving someone in need. The ferry and cross senses extend this idea to helping people move across water or a barrier.
濟 combines 氵 (water) with 齊, which likely contributes the sound and the idea of evenness or completeness. The original sense of crossing water or providing aid may derive from the idea of making a passage smooth or balanced.
Water 氵 on the left and the dense, balanced shape 齊 on the right suggest using water to smooth a crossing or to bring relief. Picture a ferryman using water to help people cross a river, providing aid and relief.
For サイ, imagine a sigh of relief when help arrives: sigh -> サイ. The ferryman's aid makes the travelers sigh with gratitude.