Car or train for Fukuoka?

I plan to visit Kyushu early oct & base out of Fukuoka. Would be bringing my mum (70+ years old) & auntie (50+ years old) who cannot walk fast or for long distances. Plan to also do a day trip to Nagasaki to see the Oct festival.

Thus, would it be better for me to hire a car instead of taking public transportation?

As you say "base out of Fukuoka" it sounds like you are going to do a lot of day trips, in that case yes, I think in your situation a car is a viable option. Car get uncomfortable in Japan if it is far distance, but that wouldn't be the case. Also while Fukuoka itself you should be prepared to pay for parking everywhere, outside of Fukuoka you might be even able to find free parking, in some of the small places.
For excursions inside Fukuoka, maybe you can think of a combination of public transport and taxi? Might still be easier and cheaper than hiring a car also for inside Fukuoka, as you would need to get parking everywhere.

Enjoy your trip with your family to Kyushu!

More info needed. Itinerary or list of locations in Fukuoka and Kyushu you want to visit.

Plan to also do a day trip to Nagasaki to see the Oct festival.

So I guess your question is about how to deal with Nagasaki Kunchi festival held 7th yo 9th October.

1. Nagasaki is very hilly city and the road is very narrow in general with trams everywhere. I am a Japanese living in Kyushu but I don't want to drive in Nagasaki city even for normal weekdays.

2. Nagasaki Kunchi is rather big festival with many locals and tourists. There are restricted areas where reserved seats are available.

http://nagasaki-kunchi.com

This is the website for the last year's festival. This year's site will open in May (maybe) but you can predict how to reserve seats etc. As you see this site is only in Japanese and there seems to be no English page. This might be because, at present, Nagasaki Kunchi is not so popular for foreign tourists. In any case, you must deal with the Japanese website to reserve your seats, prepare transportation (maybe train or bus to Nagasaki and then take taxi, but I am not sure this works for you because you may need to go up many stairs depending on the venue).

So if you really want to see this festival I would recommend you to consult with a travel agency which will arrange your transport, seat reservation and maybe accommodation.

Itinerary as follows:
6th Oct: Fukuoka
- Umina. Hanamatsu (Flower Gardens)
- Dazaifu

7th Oct: Nagasaki
- Nagasaki Kunchi (Float show)
- Chinatown
- Kofukuji Temple
- Sofukuji Temple

8th Oct: Saga
- Yutoku Inari Shrine
- Yoshinogari Park

9th Oct: Kitakyushu
- Kawachi Fujien Wisteria Garden (Garden with spectacular flower tunnels)
- Kokura Castle & Katsuyama Park

First, for Fukuoka, not sure what you are referring to. Uminonakamichi? Dazaifu is fine, though there are other better sights to see if you have time, like the Nanzoin Temple, plus if you like gardens there is Rakusuien, Ohori Koen Teien, and Yusentei.
http://www.thejapanfaq.com/fukpic.html
https://goo.gl/X3LDaE

For Nagasaki, you seem to be mostly interested in the Chinese side of the city - if so, you should definitely add the Koshibyo Shrine (and museum next to it is also very good). Very nice and often overlooked.
https://goo.gl/hZVjBi
The Kofukuji Temple is another place to consider. If you have the energy, the night view from atop Mt Inasa is one of Japan's best.

For the next day, Yutoku Inari is excellent. The Yoshinogari Park is fine, though you might study up on Yayoi period history of Japan to make it more meaningful.
https://goo.gl/dnqT25
You could possibly add more as well - the Arita Porcelain Park would be a nice addition - like walking right into Germany.

That said, you seem to be misinformed about the Kawachi Fujien - the flower season is the end of April and the first week of May. The garden is typically closed in October - it is open in spring and for the autumn colors (late Nov-early Dec).
https://www.japan-guide.com/e/e4881.html
Kokura Castle is meh, OK but not that great. Don't miss the small garden next to it. If you want to see a spectacular city view, go up to Sarakurayama just before sunset.
https://youtu.be/Hl1pCW9Ibbk