Bottom line is yes.
They are considered to be running an "enterprise" (not a business, but an enterprise) and if their turnover exceeds the threshold (might be $100k now) then they must register for GST.

Referees are exempt from paying tax as its considered a "hobby." Not sure if this applies for those officiating NBL/NBL1 but I know it to be true for those on domestic/rep games

Not a hobby for NBL refs

NBL refs often have a regular job e.g. accountant, teacher.

Curious Bigfish why do you ask?

Current rules require a business to register for GST if their turnover exceeds $75,000 per annum (for-profit businesses) or $150,000 per annum, (not-for-profit businesses). See ato.gov.au for details. Businesses with revenue under these amounts can still register for GST purposes but they don't have to.

Basketball SA revenue in 2021 was $5.6 million, so it's registered for GST purposes. The Australian Business Register confirms this (see https://www.abr.business.gov.au/ABN/View?abn=42467820914). So they'd be charging GST on ticket fees.

Similarly, if the venue operator is registered for GST purposes then they'd be including GST in the fee they charge to Basketball SA. I imagine all the Basketball SA venue operators would be registered for GST purposes.

If the referees are contractors, it'd depend on the nature of the contracting relationship between each referee and Basketball SA. This is where the 'hobby' vs 'business' issue comes in. e.g., someone being paid say $10,000+ a year ($200+ a week) for refereeing would have a hard time convincing ATO it's a hobby. If they're refereeing and invoicing Basketball SA as (part of) a business, then they'd operate under the same GST requirements as above. If they're refereeing on a hobby basis, then GST requirements wouldn't apply.

If the referees are oeprating as employees (e.g., casual employees), then there'd be no GST included in their payments from Basketball SA. Normal PAYG tax arrangements would apply.

Playing Basketball in SA District or other comps
When we pay $11 each week to play with Team Pay or other systems, does the cost include g.s.t?

Yes, as it is a private enterprise. There is a shortcut method to figuring out if an expense has GST applied to it without access to an invoice - $11 indicates they are charging $10 for the service and remitting 10% (the remaining $1) to the ATO.

For example, Basketball SA collect $176 from a game
Do they give 10% to the government for g.s.t?

Again yes and $176 indicates they are charging $160+10% GST.

Help is appreciated in understanding if this is paid or if for some reason they might be exempt

Refs then get paid, say $20 per game, the Refs are contractors so do they charge g.s.t?

No, as they aren't required to be registered for GST unless their turnover goes over $75,000 in a financial year.

The court hire is paid to the venue, say it is $60 per hour, does this attract g.s.t?

It depends on who owns the venue. If it is a local council they are exempt since the government doesn't collect GST from itself. If it were a privately owned and operated recreation centre charging the same market rate you would be slugged $66 instead.

You can use ABN Lookup to check the current GST registration status of an entity:
Teampay
https://abr.business.gov.au/ABN/View?abn=65633721383

Goods & Services Tax (GST): Registered

Basketball SA
https://abr.business.gov.au/ABN/View?abn=42467820914

Goods & Services Tax (GST): Registered

"Referees are exempt from paying tax as its considered a "hobby."

Love it, more "my mate at the pub told me" tax advice.

Any income is subject to income tax, and even semi-pros have to declare their income. But nobody is going to hunt you down for paying a teenager 20 bucks to ref your kids game.

Sorry D2.0, but LHY is correct.

Maybe don't go shooting your mouth off, or in this case keyboard, if you don't know what you're talking about.