Can't reach bank web sites

Question

I have a mystery that no one seems able to solve. I am running 10.9.5 on a 2013 MacBookAir. I have an Apple Express network that is also used by an 2011 iMac 21, also running 10.9.5. All software, including Safari, is up to date.
I have done banking on line for several years. About a month ago, I suddenly could not connect to my bank's web site home page, and thus not to the log-in portal. Neither could the iMac. However, both could log in to every other web site. First thought was that the bank site was down. I tried several times over the course of a couple of days, but no luck. I called the bank, which said their site was up.
Was the problem with my computer or my network? To find out, I took the MacBook to my office. There it could connect to the bank site, as could my office 2012 iMac 27. After searching around the bank site, I found a technical note detailing what plug-ins and other settings were required. I checked them all on the MacBook and the 27 to make sure they were the same, including updating Java to latest Mac version. I also updated the DNS server list to make sure the MacBook was running the same as the 27 (OpenDNS). 
I brought the Air back home, same lack of connection. I poked around in the wireless network list and found an unsecured network in the neighborhood. I switched to that and was able to connect; ditto with the iMac. When I switched back to my own network, no connection. I called the bank's technical support line, but IT doesn't talk to bank customers.
Everything pointed to something being wrong in my network. So I decided to re-set the Express and create a new network with a different name and password. I had high hopes for a solution. Not so. Still no connection to this one web site.  After a few days, another bank’s web site fell victim to the same problem.
After purchasing a Thunderbolt-to-Ethernet connector for $30, I was able to bypass the wireless network and plug in the Ethernet cable. Both web sites opened as they should.
The problem must be in the Express.  Does anyone know what’s going on?  Short of replacing the Express, is there a solution?

Answer 1

First step is to narrow down the proof of the Express being the cause.
Can you please connect the computer directly to the main router.
If it is a cable modem you will need to power down the modem for 15min (but test as 5min is often enough). plug the computer into it by ethernet .. and then test your bank site.. if it opens and you have no problems you have proven the issue is the Express.
If it fails then there is something else wrong.. and you are looking at the wrong area.. it has something to do with ISP routing..
Now most likely the Express is the problem.. but proof is good. Then you can be sure if you replaced the express the problem would go away.
Then I recommend you test via terminal.. the details are as follows.
Open terminal..
Type in
traceroute www.bankname.com (where you replace the bankname with the actual URL to your bank website).
See if the bankname is resolved by DNS to an IP address.. and see how far you get.
You should then test when you are connected at another location where you do get access.. is the IP the same.. is the traceroute the same..
These comparisons might not produce much but it is how you would start to break down the problem.
If it was me I would simply immediately replace the Express with a true router.. not a box that is primarily intended as AP or to use for airplay. Play being operative word here.
(If your express is Gen1 then I would reset to factory and flash back to 7.6.1 firmware) if it is gen2 and you are on 7.6.4 I think flashing back to earlier firmware has been made impossible.