I can answer this from a historical ecology perspective, but mods let me know if this strays too far from Ask Historians.

The short answer is yes, but slowly. In true biologist-boring-name fashion, this migration is known as "Lessepsian", after the modern canal’s designer, Ferdinand de Lesseps.

The canal is about 160 km long, but not all of that was freshly dug. Steam engines, blasting, railroads, all the things that make canal building easy were still in their infancy, so much of the canal was dug by hand. This was expensive, deadly, and slow, so the canal was purposely routed through several existing salt lakes to reduce the length. Additionally, the canal has no locks. On the sections that were dug, depth was kept to a minimum, usually around 10 meters.

Those are the two main barriers to migration through the canal. The center lakes were far too saline for most marine life, and the canal itself was an expanse of shallow, muddy, barren seabed. These problems have gradually been reduced over time; algae and plants have colonized the canal, the salinity has equaled out, and the canal itself has been greatly expanded over the years.

In terms of what exactly migrates through, they are generally tropical Red Sea species moving to the Mediterranean and very rarely the other way around. The Red Sea is a much more biodiverse and ecologically energic environment, more heat, more food, more life. By contrast, the Mediterranean is colder and less competitive. It’s much easier for a highly competitive species to invade a less competitive environment than vice versa. The earliest species to cross the canal were likely plankton, free floating in the water, and hardy species like crustaceans which could tolerate the desolate environment.

I’ve been perusing early British scientific journals to see if there’s any particularly early results for scientific observation of the invasive species. I’m finding scant secondhand reference to portunus pelagicus, a red sea crab and important commercial fish, being present in Port Said by 1898. Some species may also have traveled by boat hull. A flathead lobster, thennus orientalis, was first seen in Rijeka in Croatia in 1896.

The work to consult on these early migrations appears to be Charles S. Elton’s “The Ecology of Invasions by Animals and Plants”, which is sadly out of print everywhere. There are several detailed reports from a Cambridge University expedition in 1924 which reveal an abundance of interlopers by then.

More pragmatically, you can look at what fishermen now catch in the eastern Mediterranean. Penaeus japonicus, the japanese tiger prawn can be found in fishmarkets in the area.

In modern days, the ecological impact of the canal is well studied on both commercial fisheries and on broader ecological trends. Red Sea species almost uniformly out-compete Mediterranean ones, and with the salinity in the canal dropping year over year and various further plans to expand the canal, it seems likely more migration is inevitable.