First, From an economic point of view the euro has quite a few flaws. For example the fact that there is no fiscal Union makes it impossible for the central bank to react adequately to asymmetric shocks. This comes into effect if a crisis primarily appears in a few member states.
However, I believe that the export industry of especially Germany has benefitted a lot. Compared to DM the euro is worth less. This (implicit) devaluation of the currency essentially is a massive subvention to the country’s important export industry.
For the occurrence of the 2008/2008 recession itself i don’t believe in big differences. There were precedessor systems to the euro that connected the currencies in Europe.
In order to have even the slightest chance at preventing the crisis from “swapping over” from the US a Bretton Woods like system would have been required (but Bretton Woods collapsed for a reason).
However what could have changed are the options for reacting. As I’ve stated above the Euro has a hard time reacting to crisis. This could have worked better with a national currency
Also - not changing the currency would not have influenced the decisions leading to the crisis. I don’t think there would be massive butterflies regarding banking sector and regulatory oversight.