Andy took a hell of a chance walking into all the Portland banks

Yes, he did take a big chance, but Andy never was one to shy away from risks. Like when he almost got thrown off the roof when he asked the murderous guard "Do you trust your wife?"

And, as we know by watching the movie his risks usually paid off, as did his successful escape to Mexico. Andy became my hero after that!! 

..*.. TxMike ..*..
Sometimes I think we're alone in the universe, and sometimes not.

Even if someone did challenge Andy he had all the correct ID to prove he wasn't Andy Dufresne.

I believe how we the audience viewed the time from Andy's escape and his going around to various banks in the area are not meant to be taken literally. I am surmising that after the Warden realized Andy had indeed escaped beyond the prison walls he went to the safe to verify / check to make sure he still had the original ledgers / records of his illegal prison program and all the kickbacks & deposits he was making in a false name. Once the warden knew he was no longer in possession of the ledger / records book he would have remained silent about it and hoped that Andy would have been apprehended with his missing ledger / record books. Once Andy cleaned up and changed his clothes and walked into the first of many banks that day he was no longer Andy Dufresne but Randall Stevens the legal account holder and had already obtained all the proper government ID required for a new identity prior to breaking out of Shawshank prison. I suspect Andy / Randall used taxi's to reduce travel times to all the various bank locations before he fled the state. Given the fact the movie alludes to us viewers the pictures of the prison inmates are not updated like in a modern day prison. So the initial picture the authorities would have had to rely upon would have been 20+ years out of date. Also add the lack or poor to non-existent cooperation between various law enforcement jurisdictions during that time period and it would have taken longer than what was only minutes we the audience viewed on screen for Andy's escape and description to become widely known. Given just how methodical and well pre-planned Andy's escape plan was it would come as no surprise he had obtain a secondary set of government ID's and all associated documentation required for him to pass as someone else entirely before he purchased that car he used in the movie. Again to reiterate it is most likely the very last bank Andy visited that day he asked for his mail to be added to their outgoing ones as well. After that probably burned the Randall Stevens ID and other papers and switched to another alias / ID. Crossing the border into Mexico back in that time period would have been relatively easy for an average American citizen with a valid drivers license and other ID which Andy most like had.

In reality his whole escape plan was very risky from the get go. He took a huge gamble on being in the same cell for the next 19 years. Yes he was the warden's golden boy and had protection from the guards among other special privileges, but the fact he lived in that cell over 19 years was a huge stroke of luck for him, not to mention the fact the guards never checked the back of his posters not even once during inspections. Unless of course one Andy's cell was exempt from inspection as part of his special prisoner status.

Anyway Andy took a risk with that and he also took a risk attempting to crawl through a half a mile long pipe filled with shit that could have killed him. The way I see it is after his little victory celebration once he got through the pipe he would bathe thoroughly with the bar of soap then start his journey towards town. It is the middle of the night so he has a big head start. Once he got got close to town I imagine he found some sort of cover or shelter to lay low in, dry off, and put the suit on.

When the banks open Andy proceeds with business. I don't know how long it would've taken him to visit 10 banks in the Portland area, but considering he had all the right paperwork and ID probably not too long. I imagine he bought the car after his first bank visit to speed things along. As someone mentioned earlier nobody would have reason to be suspicious of a sharp looking guy in a suit named Randall Stevens. He would have had no problem discreetly spending that day finishing his business, buying a car, then heading out of town. Back then it would've taken at least a day to get his 20 year old picture out to the general public, of course today his picture would be on every television so he wouldn't have been to pull it off today.

One last thing regarding his 6'5 frame that someone previously brought up. The average height has only gone up about an inch from the 40's-50's. It would not have been that unusual to see a man of his height around.

Interesting. I’ve never considered this.