I simply can’t imagine the first few Nobel prizes being that big of a deal. I’d assume no one knew what it was. So, when and why did it start to become such a widely respected prize? Or was it really like that since inception?
Edit: grammar
The Nobel Prizes were a pretty big deal right from the beginning. They have become more famous over time, but they started big. There were multiple reasons for this:
The Nobel Prizes didn't have any significant competition: they were to be annually awarded, for a range of scientific fields and also for literature and the Nobel Peace Prize. They were also intended to be international, without regard for the nationality of the recipient (which led to criticism in Sweden). The literature and peace prizes meant that non-scientists and people not interested in science paid more attention than they would have to purely science prizes.
Alfred Nobel was well-known. He was famous for his invention of dynamite (and also gelignite and ballistite), to the point that when his brother Ludvig died in 1888, many newspapers published obituaries of Alfred. His inventions featured in many of these obituaries; one read in part: "Le marchand de la mort est mort" ("The merchant of death is dead"). These obituaries helped motivate the Nobel Prizes: Alfred did not want to be remembered as the greatest merchant of death of the 19th century. Alfred was also famous due to his wealth - his estate was worth about US$250 million in current US dollars when he died (and almost all of this went to fund the Nobel Prizes). While he wasn't as wealthy as ultra-rich Americans like Rockefeller and Vanderbilt, or even the richest Europeans (e.g., the richest German, Bertha Krupp, another "merchant of death", had a fortune of about $1.8 billion current US dollars) he was still rich enough to be famous for his wealth (his estate was worth over 1% of the Swedish GDP). That 94% of that substantial fortune went towards the prizes made the news, and helped make the Nobel Prizes widely-known before the first ones were awarded.
The prizes came with a lot of money: the first prizes awarded were worth a little over $1 million current US dollars.
All of these together meant that the Nobel Prizes were prestigious from the very beginning. As already said, they did become more famous over time. The value of the prizes fluctuated somewhat, but stayed approximately the same in Swedish kronor of the time until the late 1940s. That is, the prizes were not adjusted for inflation, and for much of the time from the 1920s to the 1940s, were worth only about a third of what they had been in 1901:
but by that time, the Nobel Prizes were already established and world-famous.