The sea level has changed little since 7,000 BP. Sea level fluctuations during the Pleistocene were larger and they were largely controlled by the amount of continental ice in the northern hemisphere. At the time of glacial maxima, the sea level was 70-130 m lower than today, due to continental ice sheets. This exposed most of the continental shelves around the continents, and created several land bridges, such as across the Bering Strait and between Papua New Guinea and Cape York Peninsula. During interglacials the sea level was higher, sometimes slightly higher than today. These changes in global mean sea level are called eustatic, they are not due to local crustal movement or sediment compression, but to changes in volume of ocean water. The crust is actually depressed under an ice sheet, so when the ice melts, the crust gradually rises. This is isostatic sea level change, which is regional. In the Hudson Bay and the Baltic Sea, the crust is still rising (and the sea level dropping) at up to 1 cm per year, in response to the melting of the Laurentide and Scandinavian ice sheets between 16,000 and 7,000 BP.
As the global sea level rose from -120m to the current level, river sediments deposited during this period became submerged and as such have been conserved well. Examination of submerged layers of sediments near the mouths of major low-latitude rivers such as the Niger, Indus, and Nile, reveals the strength of the river currents at different times in the past (1). All indicate increased monsoonal rainfall onto catchments of northern Africa, the Middle East,southern Asia, India at around 9,000 BP. This is confirmed by the results of climate models, and lake-level histories. The higher rainfall was mainly due to the migration of the summer rainbelt (zenithal/monsoonal rain) to a higher latitude after the end of the last glacial. In regions with broad continental shelves, an increase in ocean surface area due to sea level rise was important as well.
(1) Jacobs, D.K. and D.L. Schagian 1993. Climate-induced fluctuations in sea level during non-glacial times. Nature, 361, 710-2.