Formally, there are two ways to qualify for residency in another EU country: as an economically active person and as an economically non-active person. The difference is that an “economically non-active person” must prove that she has sufficient resources and health insurance not to become a burden for the social safety net in her country of residence.
You do not have an unconditional right to reside in another EU country based on your Dutch citizenship but once you do qualify, then your spouse has the right to live with you and work in your country of residence. So you have a bit of a chicken and egg problem: You need to qualify for your spouse to work but the resources that would allow you to become a economically non-active resident would presumably come mainly from your spouse's income.
In practice, your own right to reside in the country might not be a big problem as the formalities are typically minimal (as long as you don't try to apply for welfare benefits) but you might need to document it when applying for your spouse's residence card. Some countries (Spain, Portugal) do require EU citizens to get a residence card as well.
I can see three ways out of this (at least in theory, I don't have first-hand experience with this scenario):
- Use savings to show you have sufficient resources. The amounts you need should be equivalent to basic welfare benefits for one year (in countries like France or Germany, that's 300-500 € per month for an adult plus some money for accommodation and child care but it obviously varies across the EU). You would then be using your right to stay as an economically non-active person, your spouse would get a residence card and could work. Come next year, you would hopefully still have your savings because you are not actually using them to cover your living expenses, but only to qualify for the residence card.
- Get work. It does not need to be fulltime or to provide for your family, the wage might even be under the level for welfare benefits but it would help you sidestep the other requirements because the freedom of movement for workers is very extensive. Formally, the work must be “effective and genuine” but case law has established that 10-12 hours a week or a short fixed-term contract are enough. You would then use your work contract to get a residence card for you (usually not mandatory but could be useful for the next step) and your spouse (that's mandatory everywhere), who would then have the right to work as well. Formally, even merely looking for work should be enough to be considered a “worker” for at least six months but I imagine it might be difficult to sponsor a spouse on that basis.
- Have your spouse apply independently as an Australian citizen. It could be much more difficult but, depending on his or her line of work, education, etc. it might still be possible. Probably the least attractive solution, though, because visa fees and various bureaucratic requirements that cannot be imposed on EU citizens and their family would then apply.
After five years under one these statuses, the three of you would become permanent residents, your spouse would get a five-year residence card, resources conditions would not apply anymore and it would be nigh impossible to ask you to leave so that even if something would happen to you or your spouse, your family would not have to worry about that.
Note that, since you are a Dutch citizen, none of this applies to the Netherlands. In that case, Dutch law applies and I believe it's often more restrictive.