How strong was Babylon as a country during Isaiah’s time

This answer is in respect of three periods during which the Book of Isaiah was written, although Isaiah son of Amoz was certainly the original and most important author.
Isaiah, son of Amoz, also known as First Isaiah, signed that he was writing of the events he saw during the reigns of Uzziah, Jotham, Ahaz and Hezekiah. This was a time when Assyria was in the ascendancy and Babylon was only a subject state within the Assyrian Empire. So, Babylon was not strong during this time.

There was also an author, now known as Second Isaiah, who wrote more than a hundred years later during the Babylonian Exile and whose works were subsequently added to those of First Isaiah. Babylon threw off Assyrian power in 625 BCE and was at the height of its power during the period of the Babylonian Exile. Nebuchadnezzar defeated the Egyptians at Carchemish in 605, and four years later won a decisive battle against the Egyptians on the Gaza plain. Thus the only power capable of challenging Babylon had suffered two defeats by Babylon.

Third Isaiah is a further, anonymous author who wrote shortly after the Return from Exile. Persia had defeated Babylon, so once again, Babylon was an impotent subject state within a foreign empire.