How to determine the direction of a wave propagation?

For a particular section of the wave which is moving in any direction, the phase must be constant. So, if the equation says $y(x,t) = A\cos(\omega t + \beta x + \phi)$, the term inside the cosine must be constant. Hence, if time increases, $x$ must decrease to make that happen. That makes the location of the section of wave in consideration and the wave move in negative direction.

Opposite of above happens when the equation says $y(x,t) = A\cos(\omega t - \beta x + \phi)$. If t increase, $x$ must increase to make up for it. That makes a wave moving in positive direction.

The basic idea:For a moving wave, you consider a particular part of it, it moves. This means that the same $y$ would be found at other $x$ for other $t$, and if you change $t$, you need to change $x$ accordingly.

Hope that helps!

First the assumption/definition is that $\omega$ and $\beta$ are positive constants.
Next you are asking about the phase velocity ie the velocity of a crest, a trough, any fixed point on wave profile.
This means that $\omega t \pm \beta x + \phi$, which can be called the phase of the wave, is a constant.

If you differentiate $\omega t \pm \beta x + \phi=\rm constant $ With respect to time you get the component of the velocity of the wave in the x-direction $\dfrac{dx}{dt} = \mp \dfrac{\omega}{\beta}$.

So with the bracket $\omega t + \beta x + \phi$ the wave is travelling in the negative x direction (negative component) and with the bracket $\omega t - \beta x + \phi$ the wave is travelling in the positive x-direction (positive component).