Most of the time, no
To test this, I used my AnyDice.com program, invoking it for difficulties between 3 to 10 and dice pools ranging from 1 (rolling your single dot in a relative attribute) to 30 (a 3- Generation vampire rolling with some good bonus, such as 10 in Stamina and Fortitude, 5-dice worth armour, and 5 successes on Armour of Caine's Fury).
DO NOT LOOK AT THE "GRAPH" TAB -- IT WILL CRUSH YOUR BROWSER! Those 240 lines are also completely meaningless because negative numbers are not clamped at 0 and botch is indicated by -1000.
Output of -1000 indicates, again, a botch. Negative numbers and zeros are fails, positive numbers are different amounts of successes.
For difficulties 3-6, growing dice pool always means lower chance to botch/fail a roll and bigger average amount of successes and probability to get 1 success. However, for bigger difficulties there are very few actual cases of bigger dice pool meaning a bigger chance to botch.
When the answer is "yes"
1 die against any difficulty has a chance to botch of exactly 10%, for obvious reasons. Against difficulties of 7-10, some dice pools have a bigger chance to botch.
- 11% for dice pool of 2 against difficulty of 7.
- Difficulty of 8 has bigger chances for dice pools of 2, 3 and 4, namely 13%, 12.7% and 11.05%.
- Difficulty of 9 has this situations for dicepools of 2-9.
- Difficulty of 10 -- for dice pools of 2-6. For all of those difficulties the chance to botch always decreases for bigger dice pools, transition from 1 to 2 being the only exception.
So, should I just ask my Storyteller to roll 1 die for higher difficulties?
The mathematically expected amount of successes always grows with growing dicepool, no matter the difficulty. You may use this program, being almost same, except that in indicates both fails and botches as 0s, allowing you to use the "Summary" tab and look for always growing mean. Even for difficulty of 10, adding a die actually helps.
Also, regarding the chances to botch, there is no difference at all between a character with or without specialization in a given roll, because if you have rolled at least one 10, it is not a botch already. There IS a strong difference in terms of average amount of successes, thoguh, averaging at 0.1 successe per die.
Whatever you are attempting to do, always use your full dice pool. If you really can't afford to botch, you should spend Willpower. For higher difficulties it essentially adds a steady equivalent of huge amount of dice: for example, 10 dice against difficulty of 9.